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vector-toc-toggle"> Toggle Constructions subsection </button> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Products_and_sums" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Products_and_sums"> <div class="vector-toc-text"> 2.1 Products and sums </div> </a> <ul id="toc-Products_and_sums-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integer_functions_and_formal_sums" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integer_functions_and_formal_sums"> <div class="vector-toc-text"> 2.2 Integer functions and formal sums </div> </a> <ul id="toc-Integer_functions_and_formal_sums-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Presentation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Presentation"> <div class="vector-toc-text"> 2.3 Presentation </div> </a> <ul id="toc-Presentation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-As_a_module" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#As_a_module"> <div class="vector-toc-text"> 3 As a module </div> </a> <ul id="toc-As_a_module-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 4 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Universal_property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Universal_property"> <div class="vector-toc-text"> 4.1 Universal property </div> </a> <ul id="toc-Universal_property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rank" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rank"> <div class="vector-toc-text"> 4.2 Rank </div> </a> <ul id="toc-Rank-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subgroups"> <div class="vector-toc-text"> 4.3 Subgroups </div> </a> <ul id="toc-Subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Torsion_and_divisibility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Torsion_and_divisibility"> <div class="vector-toc-text"> 4.4 Torsion and divisibility </div> </a> <ul id="toc-Torsion_and_divisibility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry"> <div class="vector-toc-text"> 4.5 Symmetry </div> </a> <ul id="toc-Symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_other_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_other_groups"> <div class="vector-toc-text"> 4.6 Relation to other groups </div> </a> <ul id="toc-Relation_to_other_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 5 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Algebraic_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_topology"> <div class="vector-toc-text"> 5.1 Algebraic topology </div> </a> <ul id="toc-Algebraic_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_geometry_and_complex_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_geometry_and_complex_analysis"> <div class="vector-toc-text"> 5.2 Algebraic geometry and complex analysis </div> </a> <ul id="toc-Algebraic_geometry_and_complex_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_rings"> <div class="vector-toc-text"> 5.3 Group rings </div> </a> <ul id="toc-Group_rings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 6 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Free abelian group</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 14 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-14" aria-hidden="true" > 14 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D8%A2%D8%A8%D9%84%D9%8A%D8%A9_%D8%AD%D8%B1%D8%A9" title="زمرة آبلية حرة – Arabic" lang="ar" hreflang="ar" data-title="زمرة آبلية حرة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Freie_abelsche_Gruppe" title="Freie abelsche Gruppe – German" lang="de" hreflang="de" data-title="Freie abelsche Gruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_abeliano_libre" title="Grupo abeliano libre – Spanish" lang="es" hreflang="es" data-title="Grupo abeliano libre" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_ab%C3%A9lien_libre" title="Groupe abélien libre – French" lang="fr" hreflang="fr" data-title="Groupe abélien libre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%9C%A0_%EC%95%84%EB%B2%A8_%EA%B5%B0" title="자유 아벨 군 – Korean" lang="ko" hreflang="ko" data-title="자유 아벨 군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_abelian_bebas" title="Grup abelian bebas – Indonesian" lang="id" hreflang="id" data-title="Grup abelian bebas" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%90%D7%91%D7%9C%D7%99%D7%AA_%D7%97%D7%95%D7%A4%D7%A9%D7%99%D7%AA" title="חבורה אבלית חופשית – Hebrew" lang="he" hreflang="he" data-title="חבורה אבלית חופשית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vrije_abelse_groep" title="Vrije abelse groep – Dutch" lang="nl" hreflang="nl" data-title="Vrije abelse groep" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%94%B1%E3%82%A2%E3%83%BC%E3%83%99%E3%83%AB%E7%BE%A4" title="自由アーベル群 – Japanese" lang="ja" hreflang="ja" data-title="自由アーベル群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_abelowa_wolna" title="Grupa abelowa wolna – Polish" lang="pl" hreflang="pl" data-title="Grupa abelowa wolna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_abeliano_livre" 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Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algebra of formal sums</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Free_group" title="Free group">Free group</a>.</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a free abelian group is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> with a <a href="/wiki/Free_module" title="Free module">basis</a>. Being an abelian group means that it is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> with an addition <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> that is <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>, <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, and invertible. A basis, also called an integral basis, is a <a href="/wiki/Subset" title="Subset">subset</a> such that every element of the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> can be uniquely expressed as an <a href="/wiki/Integer" title="Integer">integer</a> <a href="/wiki/Linear_combination" title="Linear combination">combination</a> of finitely many basis elements. For instance the two-dimensional <a href="/wiki/Integer_lattice" title="Integer lattice">integer lattice</a> forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, and may equivalently be called free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">-modules, the <a href="/wiki/Free_module" title="Free module">free modules</a> over the integers. <a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice theory</a> studies free abelian <a href="/wiki/Subgroup" title="Subgroup">subgroups</a> of <a href="/wiki/Real_number" title="Real number">real</a> vector spaces. In <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, free abelian groups are used to define <a href="/wiki/Chain_(algebraic_topology)" title="Chain (algebraic topology)">chain groups</a>, and in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> they are used to define <a href="/wiki/Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisors</a>. The elements of a free abelian group with basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> may be described in several equivalent ways. These include formal sums over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, which are expressions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum a_{i}b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum a_{i}b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5e7d555a59db95749a4362009f177b8cbf7a1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.668ex; height:2.843ex;" alt="{\textstyle \sum a_{i}b_{i}}"> where each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> is a nonzero integer, each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a8c2db2990a53c683e75961826167c5adac7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.797ex; height:2.509ex;" alt="{\displaystyle b_{i}}"> is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed <a href="/wiki/Multiset" title="Multiset">multisets</a> containing finitely many elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> to the integers with finitely many nonzero values; for this functional representation, the group operation is the <a href="/wiki/Pointwise" title="Pointwise">pointwise</a> addition of functions. Every set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> has a free abelian group with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> as its basis. This group is unique in the sense that every two free abelian groups with the same basis are <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a>. Instead of constructing it by describing its individual elements, a free abelian group with basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> may be constructed as a <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a> of copies of the additive group of the integers, with one copy per member of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. Alternatively, the free abelian group with basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> may be described by a <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> with the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> as its generators and with the <a href="/wiki/Commutator" title="Commutator">commutators</a> of pairs of members as its relators. The <a href="/wiki/Rank_of_an_abelian_group" title="Rank of an abelian group">rank</a> of a free abelian group is the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a <a href="/wiki/Quotient_group" title="Quotient group">quotient</a> of a free abelian group by "relations", or as a <a href="/wiki/Cokernel" title="Cokernel">cokernel</a> of an <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> <a href="/wiki/Group_homomorphism" title="Group homomorphism">homomorphism</a> between free abelian groups. The only free abelian groups that are free groups are the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a> and the <a href="/wiki/Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_examples">Definition and examples</h2>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=1" title="Edit section: Definition and examples">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lattice_in_R2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Lattice_in_R2.svg/290px-Lattice_in_R2.svg.png" decoding="async" width="290" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Lattice_in_R2.svg/435px-Lattice_in_R2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Lattice_in_R2.svg/580px-Lattice_in_R2.svg.png 2x" data-file-width="200" data-file-height="140" /></a><figcaption>A lattice in the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>. Adding any two blue lattice points produces another lattice point; the group formed by this addition operation is a free abelian group.</figcaption></figure> A free abelian group is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> that has a basis.<a href="#cite_note-sims-1">[1]</a> Here, being an abelian group means that it is described by a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> of its elements and a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, conventionally denoted as an <a href="/wiki/Additive_group" title="Additive group">additive group</a> by the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> symbol (although it need not be the usual addition of numbers) that obey the following properties: <ul><li>The operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a> and <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>, meaning for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y=y+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y=y+x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34531a1a1d79d62f63926487d85bcd05ed2bb3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.75ex; height:2.343ex;" alt="{\displaystyle x+y=y+x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)+z=x+(y+z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)+z=x+(y+z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d53d5142d65a9596818a63fd72a130f7ad7455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.225ex; height:2.843ex;" alt="{\displaystyle (x+y)+z=x+(y+z)}">. Therefore, when combining two or more elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> using this operation, the ordering and grouping of the elements does not affect the result.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> contains an <a href="/wiki/Identity_element" title="Identity element">identity element</a> (conventionally denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">) with the property that, for every element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+0=0+x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+0=0+x=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f3ae50ede239b860fa5c3b8f078bc5e166700e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.192ex; height:2.343ex;" alt="{\displaystyle x+0=0+x=x}">.</li> <li>Every element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> has an <a href="/wiki/Inverse_element" title="Inverse element">inverse element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+(-x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+(-x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbddcc47f9ee70348043a6ba7ae80e1c9b5eb6d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.378ex; height:2.843ex;" alt="{\displaystyle x+(-x)=0}">.</li></ul> A basis is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> of the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> with the property that every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> may be formed in a unique way by choosing finitely many basis elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a8c2db2990a53c683e75961826167c5adac7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.797ex; height:2.509ex;" alt="{\displaystyle b_{i}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, choosing a nonzero integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"> for each of the chosen basis elements, and adding together <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"> copies of the basis elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a8c2db2990a53c683e75961826167c5adac7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.797ex; height:2.509ex;" alt="{\displaystyle b_{i}}"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"> is positive, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -k_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d24a6b3324d155b7ca6cc7d01567ceb08fd53f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.819ex; height:2.509ex;" alt="{\displaystyle -k_{i}}"> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -b_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -b_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90b5afba71b21d9fb6636beb51765d050d6e9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.605ex; height:2.509ex;" alt="{\displaystyle -b_{i}}"> for each basis element for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"> is negative.<a href="#cite_note-vick-2">[2]</a> As a special case, the identity element can always be formed in this way as the combination of zero basis elements, according to the usual convention for an <a href="/wiki/Empty_sum" title="Empty sum">empty sum</a>, and it must not be possible to find any other combination that represents the identity.<a href="#cite_note-3">[3]</a> The integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">, under the usual addition operation, form a free abelian group with the basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}">. The integers are commutative and associative, with <a href="/wiki/0" title="0">0</a> as the <a href="/wiki/Additive_identity" title="Additive identity">additive identity</a> and with each integer having an <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a>, its negation. Each non-negative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is the sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">, and each negative integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is the sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}"> copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">, so the basis property is also satisfied.<a href="#cite_note-sims-1">[1]</a> An example where the group operation is different from the usual addition of numbers is given by the positive <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52fdb8a7bdd61b20eae6333c26b0781cb53ea963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.319ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} ^{+}}">, which form a free abelian group with the usual <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> operation on numbers and with the <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> as their basis. Multiplication is commutative and associative, with the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> as its identity and with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55fefc6f37f48a9b4414b09ad3b17dfa739d9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle 1/x}"> as the inverse element for each positive rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">. The fact that the prime numbers forms a basis for multiplication of these numbers follows from the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, according to which every positive integer can be <a href="/wiki/Integer_factorization" title="Integer factorization">factorized</a> uniquely into the product of finitely many primes or their inverses. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=a/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=a/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611f25587778536346e4aea430ee5849ff0bcaa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.558ex; height:2.843ex;" alt="{\displaystyle q=a/b}"> is a positive rational number expressed in simplest terms, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> can be expressed as a finite combination of the primes appearing in the factorizations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. The number of copies of each prime to use in this combination is its exponent in the factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">, or the negation of its exponent in the factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">.<a href="#cite_note-fuchs-4">[4]</a> The <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> of a single variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, with integer coefficients, form a free abelian group under polynomial addition, with the powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> as a basis. As an abstract group, this is the same as (an <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> group to) the multiplicative group of positive rational numbers. One way to map these two groups to each other, showing that they are isomorphic, is to reinterpret the exponent of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">th prime number in the multiplicative group of the rationals as instead giving the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e1c39efa0100f2f99186b6dcb65a858ed461cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.23ex; height:2.676ex;" alt="{\displaystyle x^{i-1}}"> in the corresponding polynomial, or vice versa. For instance the rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5/27}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>27</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5/27}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ab0daeb407be55c8fab2f1b5c8ff28a0f33067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle 5/27}"> has exponents of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,-3,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,-3,1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f720b57af0f51b61a9ee54b52046c4c500d43689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.363ex; height:2.509ex;" alt="{\displaystyle 0,-3,1}"> for the first three prime numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,3,5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,3,5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a286d02bbc2fe6e06747240155105fab3ba348e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.555ex; height:2.509ex;" alt="{\displaystyle 2,3,5}"> and would correspond in this way to the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3x+x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3x+x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde4f371ca90ebd9735187cffbb1a282ff2cb951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.525ex; height:2.843ex;" alt="{\displaystyle -3x+x^{2}}"> having the same coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,-3,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,-3,1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f720b57af0f51b61a9ee54b52046c4c500d43689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.363ex; height:2.509ex;" alt="{\displaystyle 0,-3,1}"> for its constant, linear, and quadratic terms. Because these mappings merely reinterpret the same numbers, they define a <a href="/wiki/Bijection" title="Bijection">bijection</a> between the elements of the two groups. And because the group operation of multiplying positive rationals acts additively on the exponents of the prime numbers, in the same way that the group operation of adding polynomials acts on the coefficients of the polynomials, these maps preserve the group structure; they are <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a>. A bijective homomorphism is called an isomorphism, and its existence demonstrates that these two groups have the same properties.<a href="#cite_note-bradley-5">[5]</a> Although the representation of each group element in terms of a given basis is unique, a free abelian group has generally more than one basis, and different bases will generally result in different representations of its elements. For example, if one replaces any element of a basis by its inverse, one gets another basis. As a more elaborated example, the two-dimensional <a href="/wiki/Integer_lattice" title="Integer lattice">integer lattice</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a807ab4cb3de13a66771b5a303aca31e0391e6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{2}}">, consisting of the points in the plane with integer <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, forms a free abelian group under <a href="/wiki/Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a> with the basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(1,0),(0,1)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,0),(0,1)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5585391231b131257c6042ad554908924fcbb93b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.695ex; height:2.843ex;" alt="{\displaystyle \{(1,0),(0,1)\}}">.<a href="#cite_note-sims-1">[1]</a> For this basis, the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4,3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de9af4fe18d617acc7de162a026923a249bae30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (4,3)}"> can be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4,3)=4\cdot (1,0)+3\cdot (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4,3)=4\cdot (1,0)+3\cdot (0,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ab19d3cec8f3bb6e63e188bb8b7d37925df80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.126ex; height:2.843ex;" alt="{\displaystyle (4,3)=4\cdot (1,0)+3\cdot (0,1)}">, where 'multiplication' is defined so that, for instance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 4\cdot (1,0):=(1,0)+(1,0)+(1,0)+(1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>:=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 4\cdot (1,0):=(1,0)+(1,0)+(1,0)+(1,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e09f8f864ee952ca42d617db5ecc9dac39fd2e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.529ex; height:2.843ex;" alt="{\displaystyle \ 4\cdot (1,0):=(1,0)+(1,0)+(1,0)+(1,0)}">. There is no other way to write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4,3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de9af4fe18d617acc7de162a026923a249bae30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (4,3)}"> in the same basis. However, with a different basis such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(1,0),(1,1)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,0),(1,1)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c49c7a58af25c40d076423fd8006131f79ceea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.695ex; height:2.843ex;" alt="{\displaystyle \{(1,0),(1,1)\}}">, it can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4,3)=(1,0)+3\cdot (1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (4,3)=(1,0)+3\cdot (1,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64a1f9fd25d2765f4b15b78d7d2892d51258c685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.285ex; height:2.843ex;" alt="{\displaystyle (4,3)=(1,0)+3\cdot (1,1)}">. Generalizing this example, every <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> forms a <a href="/wiki/Finitely-generated_abelian_group" class="mw-redirect" title="Finitely-generated abelian group">finitely-generated</a> free abelian group.<a href="#cite_note-anta-6">[6]</a> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">-dimensional integer lattice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{d}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649f796a3703c840a3c78c6c6eb2e31018e5eaa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{d}}"> has a natural basis consisting of the positive integer <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a>, but it has many other bases as well: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\times d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>×</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\times d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8832c0aa14719499bb50f640be353abb7f37f069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.272ex; height:2.176ex;" alt="{\displaystyle d\times d}"> integer <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> with <a href="/wiki/Determinant" title="Determinant">determinant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}">, then the rows of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> form a basis, and <a href="/wiki/Converse_(logic)" title="Converse (logic)">conversely</a> every basis of the integer lattice has this form.<a href="#cite_note-lbr-7">[7]</a> For more on the two-dimensional case, see <a href="/wiki/Fundamental_pair_of_periods" title="Fundamental pair of periods">fundamental pair of periods</a>. <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=2" title="Edit section: Constructions">edit</a>]</div> Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by a <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation of a group</a>. <div class="mw-heading mw-heading3"><h3 id="Products_and_sums">Products and sums</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=3" title="Edit section: Products and sums">edit</a>]</div> The <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product of groups</a> consists of tuples of an element from each group in the product, with componentwise addition. The direct product of two free abelian groups is itself free abelian, with basis the <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> of the bases of the two groups.<a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;5,_p.&nbsp;75-8">[8]</a> More generally the direct product of any finite number of free abelian groups is free abelian. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">-dimensional integer lattice, for instance, is isomorphic to the direct product of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> copies of the integer group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">. The trivial group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"> is also considered to be free abelian, with basis the <a href="/wiki/Empty_set" title="Empty set">empty set</a>.<a href="#cite_note-lee-9">[9]</a> It may be interpreted as an <a href="/wiki/Empty_product" title="Empty product">empty product</a>, the direct product of zero copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">.<a href="#cite_note-trivial-product-10">[10]</a> For infinite families of free abelian groups, the direct product is not necessarily free abelian.<a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;5,_p.&nbsp;75-8">[8]</a> For instance the <a href="/wiki/Baer%E2%80%93Specker_group" title="Baer–Specker group">Baer–Specker group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{\mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82bd7cbe0951de092dc4a46c4b442b0c5d9efd70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.969ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{\mathbb {N} }}">, an <a href="/wiki/Uncountably_infinite" class="mw-redirect" title="Uncountably infinite">uncountable</a> group formed as the direct product of <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably</a> many copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">, was shown in 1937 by <a href="/wiki/Reinhold_Baer" title="Reinhold Baer">Reinhold Baer</a> to not be free abelian,<a href="#cite_note-baer-11">[11]</a> although <a href="/wiki/Ernst_Specker" title="Ernst Specker">Ernst Specker</a> <a href="/wiki/Mathematical_proof" title="Mathematical proof">proved</a> in 1950 that all of its countable subgroups are free abelian.<a href="#cite_note-specker-12">[12]</a> Instead, to obtain a free abelian group from an infinite family of groups, the <a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">direct sum</a> rather than the direct product should be used. The direct sum and direct product are the same when they are applied to finitely many groups, but differ on infinite families of groups. In the direct sum, the elements are again tuples of elements from each group, but with the restriction that all but finitely many of these elements are the identity for their group. The direct sum of infinitely many free abelian groups remains free abelian. It has a basis consisting of tuples in which all but one element is the identity, with the remaining element part of a basis for its group.<a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;5,_p.&nbsp;75-8">[8]</a> Every free abelian group may be described as a direct sum of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">, with one copy for each member of its basis.<a href="#cite_note-maclane-13">[13]</a><a href="#cite_note-kap-14">[14]</a> This construction allows any set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> to become the basis of a free abelian group.<a href="#cite_note-hungerford-15">[15]</a> <div class="mw-heading mw-heading3"><h3 id="Integer_functions_and_formal_sums">Integer functions and formal sums</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=4" title="Edit section: Integer functions and formal sums">edit</a>]</div> Given a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, one can define a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{(B)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{(B)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89545d78402ab9a606d7fc8fb9c574be5f1ce2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} ^{(B)}}"> whose elements are functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"> are two such functions, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d94a24abd865f6f9fd67a7df7e531cae1c769b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}"> is the function whose values are sums of the values in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">: that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(x)=f(x)+g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f+g)(x)=f(x)+g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf80cb50218eac1e40d4a0908bd039db3bd0863c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.795ex; height:2.843ex;" alt="{\displaystyle (f+g)(x)=f(x)+g(x)}">. This <a href="/wiki/Pointwise" title="Pointwise">pointwise</a> addition operation gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{(B)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{(B)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89545d78402ab9a606d7fc8fb9c574be5f1ce2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} ^{(B)}}"> the structure of an abelian group.<a href="#cite_note-joshi-16">[16]</a> Each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> from the given set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> corresponds to a member of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{(B)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{(B)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89545d78402ab9a606d7fc8fb9c574be5f1ce2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} ^{(B)}}">, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7347cb6483fc3e06fb296cd6df47c733e5bb0a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.256ex; height:2.009ex;" alt="{\displaystyle e_{x}}"> for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{x}(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{x}(x)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdb0079f4d43324d1acea8efa5592c31c214e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.656ex; height:2.843ex;" alt="{\displaystyle e_{x}(x)=1}"> and for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{x}(y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{x}(y)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61010bf8deb63ebb04855f3a87949d049dec791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.482ex; height:2.843ex;" alt="{\displaystyle e_{x}(y)=0}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\neq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≠</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\neq x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7b4a13fbda17b7a91e1b849576bb6dea571037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.676ex;" alt="{\displaystyle y\neq x}">. Every function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{(B)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{(B)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89545d78402ab9a606d7fc8fb9c574be5f1ce2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} ^{(B)}}"> is uniquely a linear combination of a finite number of basis elements: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\sum _{\{x\mid f(x)\neq 0\}}f(x)e_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\sum _{\{x\mid f(x)\neq 0\}}f(x)e_{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2adb5f78b91bd412603277cc61807e3fe5d2ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:20.35ex; height:6.009ex;" alt="{\displaystyle f=\sum _{\{x\mid f(x)\neq 0\}}f(x)e_{x}.}"> Thus, these elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7347cb6483fc3e06fb296cd6df47c733e5bb0a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.256ex; height:2.009ex;" alt="{\displaystyle e_{x}}"> form a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{(B)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{(B)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89545d78402ab9a606d7fc8fb9c574be5f1ce2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} ^{(B)}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{(B)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{(B)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89545d78402ab9a606d7fc8fb9c574be5f1ce2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} ^{(B)}}"> is a free abelian group. In this way, every set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> can be made into the basis of a free abelian group.<a href="#cite_note-joshi-16">[16]</a> The elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{(B)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{(B)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89545d78402ab9a606d7fc8fb9c574be5f1ce2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} ^{(B)}}"> may also be written as formal sums, expressions in the form of a sum of finitely many terms, where each term is written as the product of a nonzero integer with a distinct member of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. These expressions are considered equivalent when they have the same terms, regardless of the ordering of terms, and they may be added by forming the union of the terms, adding the integer coefficients to combine terms with the same basis element, and removing terms for which this combination produces a zero coefficient.<a href="#cite_note-fuchs-4">[4]</a> They may also be interpreted as the signed <a href="/wiki/Multiset" title="Multiset">multisets</a> of finitely many elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">.<a href="#cite_note-vggsu-17">[17]</a> <div class="mw-heading mw-heading3"><h3 id="Presentation">Presentation</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=5" title="Edit section: Presentation">edit</a>]</div> A <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation of a group</a> is a set of elements that <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generate</a> the group (meaning that all group elements can be expressed as products of finitely many generators), together with "relators", products of generators that give the identity element. The elements of a group defined in this way are <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of sequences of generators and their inverses, under an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> that allows inserting or removing any relator or generator-inverse pair as a contiguous subsequence. The free abelian group with basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> has a presentation in which the generators are the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, and the relators are the <a href="/wiki/Commutator" title="Commutator">commutators</a> of pairs of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. Here, the commutator of two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> is the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}y^{-1}xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}y^{-1}xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccfabdde6020bcc404af60de614a862a5e778e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.641ex; height:3.009ex;" alt="{\displaystyle x^{-1}y^{-1}xy}">; setting this product to the identity causes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"> to equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle yx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2e367e3fe21533f90bf7db6f03aed1d27d4176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle yx}">, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> commute. More generally, if all pairs of generators commute, then all pairs of products of generators also commute. Therefore, the group generated by this presentation is abelian, and the relators of the presentation form a minimal set of relators needed to ensure that it is abelian.<a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;3,_p.&nbsp;75-18">[18]</a> When the set of generators is finite, the presentation of a free abelian group is also finite, because there are only finitely many different commutators to include in the presentation. This fact, together with the fact that every subgroup of a free abelian group is free abelian (<a href="#Subgroups">below</a>) can be used to show that every finitely generated abelian group is finitely presented. For, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is finitely generated by a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, it is a <a href="/wiki/Quotient_group" title="Quotient group">quotient</a> of the free abelian group over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> by a free abelian subgroup, the subgroup generated by the relators of the presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">. But since this subgroup is itself free abelian, it is also finitely generated, and its basis (together with the commutators over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">) forms a finite set of relators for a presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">.<a href="#cite_note-symmetries-19">[19]</a> <div class="mw-heading mw-heading2"><h2 id="As_a_module">As a module</h2>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=6" title="Edit section: As a module">edit</a>]</div> The <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a> over the integers are defined similarly to <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> over the <a href="/wiki/Real_number" title="Real number">real numbers</a> or <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>: they consist of systems of elements that can be added to each other, with an operation for <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> by integers that is compatible with this addition operation. Every abelian group may be considered as a module over the integers, with a scalar multiplication operation defined as follows:<a href="#cite_note-algebra-20">[20]</a> <table> <tbody><tr> <td style="padding-left: 1.7em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\,x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thinmathspace" /> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\,x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/758934ab2d424901d8416c72de2a162483534021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.14ex; height:2.176ex;" alt="{\displaystyle 0\,x=0}"> </td> <td> </td> <td> </td></tr> <tr> <td style="padding-left: 1.7em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="thinmathspace" /> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\,x=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/becac8d975f3093e332d3da4dfef7e17a7b7813f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.307ex; height:2.176ex;" alt="{\displaystyle 1\,x=x}"> </td> <td> </td> <td> </td></tr> <tr> <td style="padding-left: 1.6em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,x=x+(n-1)\,x,\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\,x=x+(n-1)\,x,\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38167027b2bc40fb2c5d7b3007aad2753d680bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.273ex; height:2.843ex;" alt="{\displaystyle n\,x=x+(n-1)\,x,\quad }"> </td> <td>if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"> </td></tr> <tr> <td style="padding-left: 1.6em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,x=-((-n)\,x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mspace width="thinmathspace" /> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\,x=-((-n)\,x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79747de502d0aaffb2fc6e91903c11bb6cfecebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.203ex; height:2.843ex;" alt="{\displaystyle n\,x=-((-n)\,x),}"> </td> <td>if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9f68fb8426c4411972241aac6c359e7812eaba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n<0}"> </td></tr></tbody></table> However, unlike vector spaces, not all abelian groups have a basis, hence the special name "free" for those that do. A <a href="/wiki/Free_module" title="Free module">free module</a> is a module that can be represented as a direct sum over its base <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, so free abelian groups and free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">-modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">-module, and each free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">-module comes from a free abelian group in this way.<a href="#cite_note-ama-21">[21]</a> As well as the direct sum, another way to combine free abelian groups is to use the <a href="/wiki/Tensor_product_of_modules" title="Tensor product of modules">tensor product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">-modules. The tensor product of two free abelian groups is always free abelian, with a basis that is the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> of the bases for the two groups in the product.<a href="#cite_note-corner-22">[22]</a> Many important properties of free abelian groups may be generalized to free modules over a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>. For instance, <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodules</a> of free modules over principal ideal domains are free, a fact that <a href="#CITEREFHatcher2002">Hatcher (2002)</a> writes allows for "automatic generalization" of <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homological</a> machinery to these modules.<a href="#cite_note-hatcher-23">[23]</a> Additionally, the theorem that every <a href="/wiki/Projective_module" title="Projective module">projective</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">-module is free generalizes in the same way.<a href="#cite_note-vermani-24">[24]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=7" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Universal_property">Universal property</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=8" title="Edit section: Universal property">edit</a>]</div> A free abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> with basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> has the following <a href="/wiki/Universal_property" title="Universal property">universal property</a>: for every function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> to an abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, there exists a unique <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> which extends <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">.<a href="#cite_note-fuchs-4">[4]</a><a href="#cite_note-lee-9">[9]</a> Here, a group homomorphism is a mapping from one group to the other that is consistent with the group product law: performing a product before or after the mapping produces the same result. By a general property of universal properties, this shows that "the" abelian group of base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is unique <a href="/wiki/Up_to" title="Up to">up to</a> an isomorphism. Therefore, the universal property can be used as a definition of the free abelian group of base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. The uniqueness of the group defined by this property shows that all the other definitions are equivalent.<a href="#cite_note-hungerford-15">[15]</a> It is because of this universal property that free abelian groups are called "free": they are the <a href="/wiki/Free_object" title="Free object">free objects</a> in the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a>, the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> that has abelian groups as its objects and homomorphisms as its arrows. The map from a basis to its free abelian group is a <a href="/wiki/Functor" title="Functor">functor</a>, a structure-preserving mapping of categories, from sets to abelian groups, and is <a href="/wiki/Adjoint_functors" title="Adjoint functors">adjoint</a> to the <a href="/wiki/Forgetful_functor" title="Forgetful functor">forgetful functor</a> from abelian groups to sets.<a href="#cite_note-blass-25">[25]</a> However, a free abelian group is not a <a href="/wiki/Free_group" title="Free group">free group</a> except in two cases: a free abelian group having an empty basis (rank zero, giving the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a>) or having just one element in the basis (rank one, giving the <a href="/wiki/Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a>).<a href="#cite_note-lee-9">[9]</a><a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;4,_p.&nbsp;75-26">[26]</a> Other abelian groups are not free groups because in free groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"> must be different from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ba}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ba}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7da31f1e7472a309d2d5179ebe881dedc5f68ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ba}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> are different elements of the basis, while in free abelian groups the two products must be identical for all pairs of elements. In the general <a href="/wiki/Category_of_groups" title="Category of groups">category of groups</a>, it is an added constraint to demand that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab=ba}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab=ba}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794f6c310259e816eed4a00262d91bf4f53e37c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.553ex; height:2.176ex;" alt="{\displaystyle ab=ba}">, whereas this is a necessary property in the category of abelian groups.<a href="#cite_note-FOOTNOTEHungerford197470-27">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Rank">Rank</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=9" title="Edit section: Rank">edit</a>]</div> Every two bases of the same free abelian group have the same <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>, so the cardinality of a basis forms an <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> of the group known as its rank.<a href="#cite_note-FOOTNOTEHungerford1974Theorem&nbsp;1.2,_p.&nbsp;73-28">[28]</a><a href="#cite_note-hm-29">[29]</a> Two free abelian groups are isomorphic if and only if they have the same rank.<a href="#cite_note-fuchs-4">[4]</a> A free abelian group is <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated</a> if and only if its rank is a finite number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, in which case the group is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5de7ced4588982b574fe19894aec6a3ca4c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}">.<a href="#cite_note-machi-30">[30]</a> This notion of rank can be generalized, from free abelian groups to abelian groups that are not necessarily free. The <a href="/wiki/Rank_of_an_abelian_group" title="Rank of an abelian group">rank of an abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is defined as the rank of a free abelian subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> for which the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10a8a959da2e7dd5e66f795ff48780ba4ab53549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.73ex; height:2.843ex;" alt="{\displaystyle G/F}"> is a <a href="/wiki/Torsion_group" title="Torsion group">torsion group</a>. Equivalently, it is the cardinality of a <a href="/wiki/Maximal_element" class="mw-redirect" title="Maximal element">maximal</a> subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> that generates a free subgroup. The rank is a group invariant: it does not depend on the choice of the subgroup.<a href="#cite_note-rotman-31">[31]</a> <div class="mw-heading mw-heading3"><h3 id="Subgroups">Subgroups</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=10" title="Edit section: Subgroups">edit</a>]</div> Every subgroup of a free abelian group is itself a free abelian group. This result of <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a><a href="#cite_note-johnson-32">[32]</a> was a precursor to the analogous <a href="/wiki/Nielsen%E2%80%93Schreier_theorem" title="Nielsen–Schreier theorem">Nielsen–Schreier theorem</a> that every subgroup of a <a href="/wiki/Free_group" title="Free group">free group</a> is free, and is a generalization of the fact that <a href="/wiki/Subgroups_of_cyclic_groups" title="Subgroups of cyclic groups">every nontrivial subgroup of the infinite cyclic group is infinite cyclic</a>. The proof needs the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>.<a href="#cite_note-blass-25">[25]</a> A proof using <a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a> (one of many equivalent assumptions to the axiom of choice) can be found in <a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a>'s Algebra.<a href="#cite_note-lang-33">[33]</a> <a href="/wiki/Solomon_Lefschetz" title="Solomon Lefschetz">Solomon Lefschetz</a> and <a href="/wiki/Irving_Kaplansky" title="Irving Kaplansky">Irving Kaplansky</a> argue that using the <a href="/wiki/Well-ordering_principle" title="Well-ordering principle">well-ordering principle</a> in place of Zorn's lemma leads to a more intuitive proof.<a href="#cite_note-kap-14">[14]</a> In the case of finitely generated free abelian groups, the proof is easier, does not need the axiom of choice, and leads to a more precise result. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is a subgroup of a finitely generated free abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is free and there exists a basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{1},\ldots ,e_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{1},\ldots ,e_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12c4e5513493af62a4753b821ba5bc7842a5278a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.427ex; height:2.843ex;" alt="{\displaystyle (e_{1},\ldots ,e_{n})}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}|d_{2}|\ldots |d_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}|d_{2}|\ldots |d_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/833494b30ae334f06c0484460bbc7d4491b8f624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.262ex; height:2.843ex;" alt="{\displaystyle d_{1}|d_{2}|\ldots |d_{k}}"> (that is, each one divides the next one) such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d_{1}e_{1},\ldots ,d_{k}e_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d_{1}e_{1},\ldots ,d_{k}e_{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc879a3c7c6454281f5e674e5428d0e4b09f858a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.858ex; height:2.843ex;" alt="{\displaystyle (d_{1}e_{1},\ldots ,d_{k}e_{k})}"> is a basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}"> Moreover, the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1},d_{2},\ldots ,d_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1},d_{2},\ldots ,d_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd158eddf8c571d4b250bcf77327a0fdcb5e6273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.036ex; height:2.509ex;" alt="{\displaystyle d_{1},d_{2},\ldots ,d_{k}}"> depends only on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> and not on the basis.<a href="#cite_note-FOOTNOTEHungerford1974Theorem&nbsp;1.6,_p.&nbsp;74-34">[34]</a> A <a href="/wiki/Constructive_proof" title="Constructive proof">constructive proof</a> of the existence part of the theorem is provided by any algorithm computing the <a href="/wiki/Smith_normal_form" title="Smith normal form">Smith normal form</a> of a matrix of integers.<a href="#cite_note-FOOTNOTEJohnson200171–72-35">[35]</a> Uniqueness follows from the fact that, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\leq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≤</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\leq k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a19f094684d5866bbe1c1afe2c6bdcea8dc2aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.358ex; height:2.343ex;" alt="{\displaystyle r\leq k}">, the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of the <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a> of rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> of the matrix is not changed during the Smith normal form computation and is the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}\cdots d_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}\cdots d_{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec63aa426a6f6aad633102281f3834045dc61ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.943ex; height:2.509ex;" alt="{\displaystyle d_{1}\cdots d_{r}}"> at the end of the computation.<a href="#cite_note-norman-36">[36]</a> <div class="mw-heading mw-heading3"><h3 id="Torsion_and_divisibility">Torsion and divisibility</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=11" title="Edit section: Torsion and divisibility">edit</a>]</div> All free abelian groups are <a href="/wiki/Torsion_(algebra)" title="Torsion (algebra)">torsion-free</a>, meaning that there is no non-identity group element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and nonzero integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nx=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd41706fc0e253d4007553370d72cc191b431b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.985ex; height:2.176ex;" alt="{\displaystyle nx=0}">. Conversely, all finitely generated torsion-free abelian groups are free abelian.<a href="#cite_note-lee-9">[9]</a><a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;9,_p.&nbsp;75-37">[37]</a> The additive group of rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> provides an example of a torsion-free (but not finitely generated) abelian group that is not free abelian.<a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;10,_p.&nbsp;75-38">[38]</a> One reason that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> is not free abelian is that it is <a href="/wiki/Divisible_group" title="Divisible group">divisible</a>, meaning that, for every element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae13dca0f57812b8bb2eae6a3bb0a00321ba46a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.978ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {Q} }"> and every nonzero integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, it is possible to express <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> as a scalar multiple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ny}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ny}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9525f353caa23b1273163429dbd8f39c6d82d115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.55ex; height:2.009ex;" alt="{\displaystyle ny}"> of another element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2cd9fa0716decb3e02b377d70d704da7eb24cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.141ex; height:2.843ex;" alt="{\displaystyle y=x/n}">. In contrast, non-trivial free abelian groups are never divisible, because in a free abelian group the basis elements cannot be expressed as multiples of other elements.<a href="#cite_note-FOOTNOTEHungerford1974Exercise&nbsp;4,_p.&nbsp;198-39">[39]</a> <div class="mw-heading mw-heading3"><h3 id="Symmetry">Symmetry</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=12" title="Edit section: Symmetry">edit</a>]</div> The symmetries of any group can be described as <a href="/wiki/Group_automorphism" class="mw-redirect" title="Group automorphism">group automorphisms</a>, the <a href="/wiki/Invertible_function" class="mw-redirect" title="Invertible function">invertible</a> homomorphisms from the group to itself. In non-abelian groups these are further subdivided into <a href="/wiki/Inner_automorphism" title="Inner automorphism">inner</a> and <a href="/wiki/Outer_automorphism" class="mw-redirect" title="Outer automorphism">outer</a> automorphisms, but in abelian groups all non-identity automorphisms are outer. They form another group, the <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of the given group, under the operation of <a href="/wiki/Function_composition" title="Function composition">composition</a>. The automorphism group of a free abelian group of finite rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a331e7378914bbd230b51b583f10d7db8ecf114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.065ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {Z} )}">, which can be described concretely (for a specific basis of the free automorphism group) as the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> invertible integer matrices under the operation of <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. Their <a href="/wiki/Group_action" title="Group action">action</a> as symmetries on the free abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5de7ced4588982b574fe19894aec6a3ca4c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.769ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} ^{n}}"> is just matrix-vector multiplication.<a href="#cite_note-bridvog-40">[40]</a> The automorphism groups of two infinite-rank free abelian groups have the same <a href="/wiki/First-order_theory" class="mw-redirect" title="First-order theory">first-order theories</a> as each other, if and only if their ranks are equivalent <a href="/wiki/Cardinal_number" title="Cardinal number">cardinals</a> from the point of view of <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a>. This result depends on the structure of <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involutions</a> of free abelian groups, the automorphisms that are their own inverse. Given a basis for a free abelian group, one can find involutions that map any set of disjoint pairs of basis elements to each other, or that negate any chosen subset of basis elements, leaving the other basis elements fixed. Conversely, for every involution of a free abelian group, one can find a basis of the group for which all basis elements are swapped in pairs, negated, or left unchanged by the involution.<a href="#cite_note-tolstykh-41">[41]</a> <div class="mw-heading mw-heading3"><h3 id="Relation_to_other_groups">Relation to other groups</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=13" title="Edit section: Relation to other groups">edit</a>]</div> If a free abelian group is a quotient of two groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A/B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f07c45a0f3de042f2d4eece4339359ffd6d78f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.67ex; height:2.843ex;" alt="{\displaystyle A/B}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the direct sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\oplus A/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊕</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\oplus A/B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e43ae5001233c559f829d3ac90e9006eee358fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.274ex; height:2.843ex;" alt="{\displaystyle B\oplus A/B}">.<a href="#cite_note-fuchs-4">[4]</a> Given an arbitrary abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, there always exists a free abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and a <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> group homomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. One way of constructing a surjection onto a given group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is to let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\mathbb {Z} ^{(A)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\mathbb {Z} ^{(A)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6ab90098d3c7d2573d7dca92bdb36a789fe628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.134ex; height:2.843ex;" alt="{\displaystyle F=\mathbb {Z} ^{(A)}}"> be the free abelian group over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, represented as formal sums. Then a surjection can be defined by mapping formal sums in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> to the corresponding sums of members of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. That is, the surjection maps <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{\{x\mid a_{x}\neq 0\}}a_{x}e_{x}\mapsto \sum _{\{x\mid a_{x}\neq 0\}}a_{x}x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{\{x\mid a_{x}\neq 0\}}a_{x}e_{x}\mapsto \sum _{\{x\mid a_{x}\neq 0\}}a_{x}x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da4318e1732cf906b88416cce8af6e70bfb9eddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.304ex; height:6.009ex;" alt="{\displaystyle \sum _{\{x\mid a_{x}\neq 0\}}a_{x}e_{x}\mapsto \sum _{\{x\mid a_{x}\neq 0\}}a_{x}x,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339ff13ca52000e5467b829dfd008f6846820b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.402ex; height:2.009ex;" alt="{\displaystyle a_{x}}"> is the integer coefficient of basis element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7347cb6483fc3e06fb296cd6df47c733e5bb0a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.256ex; height:2.009ex;" alt="{\displaystyle e_{x}}"> in a given formal sum, the first sum is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}">, and the second sum is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.<a href="#cite_note-hm-29">[29]</a><a href="#cite_note-FOOTNOTEHungerford1974Theorem&nbsp;1.4,_p.&nbsp;74-42">[42]</a> This surjection is the unique group homomorphism which extends the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{x}\mapsto x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">↦</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{x}\mapsto x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2430b3c35439619e05bd54fb26782e87c3c081fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.2ex; height:2.176ex;" alt="{\displaystyle e_{x}\mapsto x}">, and so its construction can be seen as an instance of the universal property. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> are as above, the <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> of the surjection from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is also free abelian, as it is a subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> (the subgroup of elements mapped to the identity). Therefore, these groups form a <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to G\to F\to A\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to G\to F\to A\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c3177b22848f065a5831621c9618dc338e48ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.092ex; height:2.176ex;" alt="{\displaystyle 0\to G\to F\to A\to 0}"> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> are both free abelian and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is isomorphic to the <a href="/wiki/Factor_group" class="mw-redirect" title="Factor group">factor group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F/G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F/G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e811c1706b0ba39af5a55cb790c066c3e7f77273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.73ex; height:2.843ex;" alt="{\displaystyle F/G}">. This is a <a href="/wiki/Free_resolution" class="mw-redirect" title="Free resolution">free resolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.<a href="#cite_note-vick-2">[2]</a> Furthermore, assuming the axiom of choice,<a href="#cite_note-projective-choice-43">[43]</a> the free abelian groups are precisely the <a href="/wiki/Projective_module" title="Projective module">projective objects</a> in the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a>.<a href="#cite_note-fuchs-4">[4]</a><a href="#cite_note-griffith-44">[44]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=14" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Algebraic_topology">Algebraic topology</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=15" title="Edit section: Algebraic topology">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chain_(algebraic_topology)" title="Chain (algebraic topology)">Chain (algebraic topology)</a></div> In <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, a formal sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-dimensional <a href="/wiki/Simplex" title="Simplex">simplices</a> is called a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-chain, and the free abelian group having a collection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-simplices as its basis is called a chain group.<a href="#cite_note-dctm-45">[45]</a> The simplices are generally taken from some <a href="/wiki/Topological_space" title="Topological space">topological space</a>, for instance as the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-simplices in a <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a>, or the set of <a href="/wiki/Singular_homology" title="Singular homology">singular</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-simplices in a <a href="/wiki/Manifold" title="Manifold">manifold</a>. Any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-dimensional simplex has a boundary that can be represented as a formal sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e69f74fa2adbbab50f6969acb2af719045435461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k-1)}">-dimensional simplices, and the universal property of free abelian groups allows this boundary operator to be extended to a group homomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-chains to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e69f74fa2adbbab50f6969acb2af719045435461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k-1)}">-chains. The system of chain groups linked by boundary operators in this way forms a <a href="/wiki/Chain_complex" title="Chain complex">chain complex</a>, and the study of chain complexes forms the basis of <a href="/wiki/Homology_theory" class="mw-redirect" title="Homology theory">homology theory</a>.<a href="#cite_note-edelsbrunner-46">[46]</a> <div class="mw-heading mw-heading3"><h3 id="Algebraic_geometry_and_complex_analysis">Algebraic geometry and complex analysis</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=16" title="Edit section: Algebraic geometry and complex analysis">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">Divisor (algebraic geometry)</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Z4_over_z4minus1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Z4_over_z4minus1.jpg/290px-Z4_over_z4minus1.jpg" decoding="async" width="290" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Z4_over_z4minus1.jpg/435px-Z4_over_z4minus1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Z4_over_z4minus1.jpg/580px-Z4_over_z4minus1.jpg 2x" data-file-width="855" data-file-height="855" /></a><figcaption>The <a href="/wiki/Rational_function" title="Rational function">rational function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{4}/(z^{4}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{4}/(z^{4}-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd09a165130509398f44720882f1df833431b721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.264ex; height:3.176ex;" alt="{\displaystyle z^{4}/(z^{4}-1)}"> has a zero of order four at 0 (the black point at the center of the plot), and <a href="/wiki/Zeros_and_poles#Definitions" title="Zeros and poles">simple poles</a> at the four complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b7df63745bc6839de7b7df413c192f5816ff2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.611ex; height:2.176ex;" alt="{\displaystyle \pm i}"> (the white points at the ends of the four petals). It can be represented (up to a scalar) by the divisor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4e_{0}-e_{1}-e_{-1}-e_{i}-e_{-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4e_{0}-e_{1}-e_{-1}-e_{i}-e_{-i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab86f9afb561ac69d26db7a7b13f675e458efac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.261ex; height:2.509ex;" alt="{\displaystyle 4e_{0}-e_{1}-e_{-1}-e_{i}-e_{-i}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237434415c6fd3f47c255a2a07017f94affce732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.085ex; height:2.009ex;" alt="{\displaystyle e_{z}}"> is the basis element for a complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> in a free abelian group over the complex numbers.</figcaption></figure> Every <a href="/wiki/Rational_function" title="Rational function">rational function</a> over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> can be associated with a signed multiset of complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}">, the <a href="/wiki/Zeros_and_poles" title="Zeros and poles">zeros and poles</a> of the function (points where its value is zero or infinite). The multiplicity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}"> of a point in this multiset is its order as a zero of the function, or the negation of its order as a pole. Then the function itself can be recovered from this data, up to a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> factor, as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(q)=\prod (q-c_{i})^{m_{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∏</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(q)=\prod (q-c_{i})^{m_{i}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ba640116c51f855a44b4358b7b91623c634963" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.698ex; height:3.843ex;" alt="{\displaystyle f(q)=\prod (q-c_{i})^{m_{i}}.}"> If these multisets are interpreted as members of a free abelian group over the complex numbers, then the product or quotient of two rational functions corresponds to the sum or difference of two group members. Thus, the multiplicative group of rational functions can be factored into the multiplicative group of complex numbers (the associated scalar factors for each function) and the free abelian group over the complex numbers. The rational functions that have a nonzero limiting value at infinity (the <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic functions</a> on the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>) form a subgroup of this group in which the sum of the multiplicities is zero.<a href="#cite_note-dw-47">[47]</a> This construction has been generalized, in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, to the notion of a <a href="/wiki/Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisor</a>. There are different definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a>, the set of solution points of a <a href="/wiki/System_of_polynomial_equations" title="System of polynomial equations">system of polynomial equations</a>. In the case where the system of equations has one degree of freedom (its solutions form an <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curve</a> or <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a>), a subvariety has codimension one when it consists of isolated points, and in this case a divisor is again a signed multiset of points from the variety.<a href="#cite_note-acrs-48">[48]</a> The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors form a subgroup of a free abelian group over the points of the surface, with multiplication or division of functions corresponding to addition or subtraction of group elements. To be a divisor, an element of the free abelian group must have multiplicities summing to zero, and meet certain additional constraints depending on the surface.<a href="#cite_note-dw-47">[47]</a> <div class="mw-heading mw-heading3"><h3 id="Group_rings">Group rings</h3>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=17" title="Edit section: Group rings">edit</a>]</div> The <a href="/wiki/Integer" title="Integer">integral</a> <a href="/wiki/Group_ring" title="Group ring">group ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [G]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [G]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40260c366fc309a5872899d2ea34cf094855857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.671ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [G]}">, for any group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">, is a ring whose additive group is the free abelian group over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">.<a href="#cite_note-stein-szabo-49">[49]</a> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is <a href="/wiki/Finite_group" title="Finite group">finite</a> and abelian, the multiplicative group of <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">units</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [G]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [G]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40260c366fc309a5872899d2ea34cf094855857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.671ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [G]}"> has the structure of a direct product of a finite group and a finitely generated free abelian group.<a href="#cite_note-higman-50">[50]</a><a href="#cite_note-ayoub-ayoub-51">[51]</a> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Free_abelian_group&action=edit&section=18" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-sims-1">^ <a href="#cite_ref-sims_1-0">a</a> <a href="#cite_ref-sims_1-1">b</a> <a href="#cite_ref-sims_1-2">c</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSims1994" class="citation cs2">Sims, Charles C. (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=k6joymrqQqMC&pg=PA320">"Section 8.1: Free abelian groups"</a>, Computation with Finitely Presented Groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, p. 320, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511574702">10.1017/CBO9780511574702</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-43213-8" title="Special:BookSources/0-521-43213-8"><bdi>0-521-43213-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1267733">1267733</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+8.1%3A+Free+abelian+groups&rft.btitle=Computation+with+Finitely+Presented+Groups&rft.series=Encyclopedia+of+Mathematics+and+its+Applications&rft.pages=320&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1267733%23id-name%3DMR&rft_id=info%3Adoi%2F10.1017%2FCBO9780511574702&rft.isbn=0-521-43213-8&rft.aulast=Sims&rft.aufirst=Charles+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dk6joymrqQqMC%26pg%3DPA320&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-vick-2">^ <a href="#cite_ref-vick_2-0">a</a> <a href="#cite_ref-vick_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVick1994" class="citation cs2">Vick, James W. (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5Bq8YlLrNc8C&pg=PA70">Homology Theory: An Introduction to Algebraic Topology</a>, Graduate Texts in Mathematics, vol. 145, Springer, pp. 4, 70, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387941264" title="Special:BookSources/9780387941264"><bdi>9780387941264</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Homology+Theory%3A+An+Introduction+to+Algebraic+Topology&rft.series=Graduate+Texts+in+Mathematics&rft.pages=4%2C+70&rft.pub=Springer&rft.date=1994&rft.isbn=9780387941264&rft.aulast=Vick&rft.aufirst=James+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5Bq8YlLrNc8C%26pg%3DPA70&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> Some sources define free abelian groups by the condition that the only representation of the identity is the empty sum, rather than treating it as a special case of unique representation of all group elements; see, e.g., <a href="#CITEREFSims1994">Sims (1994)</a>. </li> <li id="cite_note-fuchs-4">^ <a href="#cite_ref-fuchs_4-0">a</a> <a href="#cite_ref-fuchs_4-1">b</a> <a href="#cite_ref-fuchs_4-2">c</a> <a href="#cite_ref-fuchs_4-3">d</a> <a href="#cite_ref-fuchs_4-4">e</a> <a href="#cite_ref-fuchs_4-5">f</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuchs2015" class="citation cs2"><a href="/wiki/L%C3%A1szl%C3%B3_Fuchs" title="László Fuchs">Fuchs, László</a> (2015), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2KMvCwAAQBAJ&pg=PA75">"Section 3.1: Freeness and projectivity"</a>, Abelian Groups, Springer Monographs in Mathematics, Cham: Springer, pp. 75–80, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-19422-6">10.1007/978-3-319-19422-6</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-19421-9" title="Special:BookSources/978-3-319-19421-9"><bdi>978-3-319-19421-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3467030">3467030</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+3.1%3A+Freeness+and+projectivity&rft.btitle=Abelian+Groups&rft.place=Cham&rft.series=Springer+Monographs+in+Mathematics&rft.pages=75-80&rft.pub=Springer&rft.date=2015&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3467030%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-319-19422-6&rft.isbn=978-3-319-19421-9&rft.aulast=Fuchs&rft.aufirst=L%C3%A1szl%C3%B3&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2KMvCwAAQBAJ%26pg%3DPA75&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-bradley-5"><a href="#cite_ref-bradley_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradley2005" class="citation cs2">Bradley, David M. (2005), Counting the positive rationals: A brief survey, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0509025">math/0509025</a>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005math......9025B">2005math......9025B</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Counting+the+positive+rationals%3A+A+brief+survey&rft.date=2005&rft_id=info%3Aarxiv%2Fmath%2F0509025&rft_id=info%3Abibcode%2F2005math......9025B&rft.aulast=Bradley&rft.aufirst=David+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-anta-6"><a href="#cite_ref-anta_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMollin2011" class="citation cs2">Mollin, Richard A. (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6I1setlljDYC&pg=PA182">Advanced Number Theory with Applications</a>, CRC Press, p. 182, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781420083293" title="Special:BookSources/9781420083293"><bdi>9781420083293</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Number+Theory+with+Applications&rft.pages=182&rft.pub=CRC+Press&rft.date=2011&rft.isbn=9781420083293&rft.aulast=Mollin&rft.aufirst=Richard+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6I1setlljDYC%26pg%3DPA182&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-lbr-7"><a href="#cite_ref-lbr_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBremner2011" class="citation cs2">Bremner, Murray R. (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i5AkDxkrjPcC&pg=PA6">Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications</a>, CRC Press, p. 6, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781439807026" title="Special:BookSources/9781439807026"><bdi>9781439807026</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lattice+Basis+Reduction%3A+An+Introduction+to+the+LLL+Algorithm+and+Its+Applications&rft.pages=6&rft.pub=CRC+Press&rft.date=2011&rft.isbn=9781439807026&rft.aulast=Bremner&rft.aufirst=Murray+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Di5AkDxkrjPcC%26pg%3DPA6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHungerford1974Exercise&nbsp;5,_p.&nbsp;75-8">^ <a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;5,_p.&nbsp;75_8-0">a</a> <a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;5,_p.&nbsp;75_8-1">b</a> <a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;5,_p.&nbsp;75_8-2">c</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Exercise 5, p. 75. </li> <li id="cite_note-lee-9">^ <a href="#cite_ref-lee_9-0">a</a> <a href="#cite_ref-lee_9-1">b</a> <a href="#cite_ref-lee_9-2">c</a> <a href="#cite_ref-lee_9-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2010" class="citation cs2"><a href="/wiki/John_M._Lee" title="John M. Lee">Lee, John M.</a> (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZQVGAAAAQBAJ&pg=PA244">"Free Abelian Groups"</a>, Introduction to Topological Manifolds, Graduate Texts in Mathematics, vol. 202 (2nd ed.), Springer, pp. 244–248, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781441979407" title="Special:BookSources/9781441979407"><bdi>9781441979407</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Free+Abelian+Groups&rft.btitle=Introduction+to+Topological+Manifolds&rft.series=Graduate+Texts+in+Mathematics&rft.pages=244-248&rft.edition=2nd&rft.pub=Springer&rft.date=2010&rft.isbn=9781441979407&rft.aulast=Lee&rft.aufirst=John+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZQVGAAAAQBAJ%26pg%3DPA244&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-trivial-product-10"><a href="#cite_ref-trivial-product_10-0">^</a> As stated explicitly, for instance, in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartleyTurull1994" class="citation cs2">Hartley, Brian; Turull, Alexandre (1994), "On characters of coprime operator groups and the Glauberman character correspondence", <a href="/wiki/Crelle%27s_Journal" title="Crelle's Journal">Journal für die Reine und Angewandte Mathematik</a>, 1994 (451): 175–219, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1994.451.175">10.1515/crll.1994.451.175</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1277300">1277300</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118116330">118116330</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=On+characters+of+coprime+operator+groups+and+the+Glauberman+character+correspondence&rft.volume=1994&rft.issue=451&rft.pages=175-219&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1277300%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118116330%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1515%2Fcrll.1994.451.175&rft.aulast=Hartley&rft.aufirst=Brian&rft.au=Turull%2C+Alexandre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988">, proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups" </li> <li id="cite_note-baer-11"><a href="#cite_ref-baer_11-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaer1937" class="citation cs2"><a href="/wiki/Reinhold_Baer" title="Reinhold Baer">Baer, Reinhold</a> (1937), "Abelian groups without elements of finite order", Duke Mathematical Journal, 3 (1): 68–122, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-37-00308-9">10.1215/S0012-7094-37-00308-9</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F100591">10338.dmlcz/100591</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1545974">1545974</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=Abelian+groups+without+elements+of+finite+order&rft.volume=3&rft.issue=1&rft.pages=68-122&rft.date=1937&rft_id=info%3Ahdl%2F10338.dmlcz%2F100591&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1545974%23id-name%3DMR&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-37-00308-9&rft.aulast=Baer&rft.aufirst=Reinhold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-specker-12"><a href="#cite_ref-specker_12-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpecker1950" class="citation cs2"><a href="/wiki/Ernst_Specker" title="Ernst Specker">Specker, Ernst</a> (1950), "Additive Gruppen von Folgen ganzer Zahlen", Portugaliae Math., 9: 131–140, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0039719">0039719</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Portugaliae+Math.&rft.atitle=Additive+Gruppen+von+Folgen+ganzer+Zahlen&rft.volume=9&rft.pages=131-140&rft.date=1950&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0039719%23id-name%3DMR&rft.aulast=Specker&rft.aufirst=Ernst&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-maclane-13"><a href="#cite_ref-maclane_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_Lane1995" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pxRlrJn-WPgC&pg=PA93">Homology</a>, Classics in Mathematics, Springer, p. 93, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540586623" title="Special:BookSources/9783540586623"><bdi>9783540586623</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Homology&rft.series=Classics+in+Mathematics&rft.pages=93&rft.pub=Springer&rft.date=1995&rft.isbn=9783540586623&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpxRlrJn-WPgC%26pg%3DPA93&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-kap-14">^ <a href="#cite_ref-kap_14-0">a</a> <a href="#cite_ref-kap_14-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaplansky2001" class="citation cs2"><a href="/wiki/Irving_Kaplansky" title="Irving Kaplansky">Kaplansky, Irving</a> (2001), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1XFDM75VK5MC&pg=PA124">Set Theory and Metric Spaces</a>, AMS Chelsea Publishing Series, vol. 298, American Mathematical Society, pp. 124–125, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821826942" title="Special:BookSources/9780821826942"><bdi>9780821826942</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Metric+Spaces&rft.series=AMS+Chelsea+Publishing+Series&rft.pages=124-125&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=9780821826942&rft.aulast=Kaplansky&rft.aufirst=Irving&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1XFDM75VK5MC%26pg%3DPA124&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-hungerford-15">^ <a href="#cite_ref-hungerford_15-0">a</a> <a href="#cite_ref-hungerford_15-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHungerford1974" class="citation cs2"><a href="/wiki/Thomas_W._Hungerford" title="Thomas W. Hungerford">Hungerford, Thomas W.</a> (1974), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=t6N_tOQhafoC&pg=PA70">"II.1 Free abelian groups"</a>, Algebra, Graduate Texts in Mathematics, vol. 73, Springer, pp. 70–75, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387905181" title="Special:BookSources/9780387905181"><bdi>9780387905181</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=II.1+Free+abelian+groups&rft.btitle=Algebra&rft.series=Graduate+Texts+in+Mathematics&rft.pages=70-75&rft.pub=Springer&rft.date=1974&rft.isbn=9780387905181&rft.aulast=Hungerford&rft.aufirst=Thomas+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dt6N_tOQhafoC%26pg%3DPA70&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988">. See in particular Theorem 1.1, pp. 72–73, and the remarks following it. </li> <li id="cite_note-joshi-16">^ <a href="#cite_ref-joshi_16-0">a</a> <a href="#cite_ref-joshi_16-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoshi1997" class="citation cs2">Joshi, K. D. (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lxIgGGJXacoC&pg=PA45">Applied Discrete Structures</a>, New Age International, pp. 45–46, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788122408263" title="Special:BookSources/9788122408263"><bdi>9788122408263</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Discrete+Structures&rft.pages=45-46&rft.pub=New+Age+International&rft.date=1997&rft.isbn=9788122408263&rft.aulast=Joshi&rft.aufirst=K.+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlxIgGGJXacoC%26pg%3DPA45&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-vggsu-17"><a href="#cite_ref-vggsu_17-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_GlabbeekGoltzSchicke-Uffmann2013" class="citation cs2">van Glabbeek, Rob; <a href="/wiki/Ursula_Goltz" title="Ursula Goltz">Goltz, Ursula</a>; Schicke-Uffmann, Jens-Wolfhard (2013), "On characterising distributability", Logical Methods in Computer Science, 9 (3): 3:17, 58, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1309.3883">1309.3883</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2168%2FLMCS-9%283%3A17%292013">10.2168/LMCS-9(3:17)2013</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3109601">3109601</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17046529">17046529</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Logical+Methods+in+Computer+Science&rft.atitle=On+characterising+distributability&rft.volume=9&rft.issue=3&rft.pages=3%3A17%2C+58&rft.date=2013&rft_id=info%3Aarxiv%2F1309.3883&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3109601%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17046529%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2168%2FLMCS-9%283%3A17%292013&rft.aulast=van+Glabbeek&rft.aufirst=Rob&rft.au=Goltz%2C+Ursula&rft.au=Schicke-Uffmann%2C+Jens-Wolfhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHungerford1974Exercise&nbsp;3,_p.&nbsp;75-18"><a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;3,_p.&nbsp;75_18-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Exercise 3, p. 75. </li> <li id="cite_note-symmetries-19"><a href="#cite_ref-symmetries_19-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson2001" class="citation cs2">Johnson, D. L. (2001), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BecLeCWOjI4C&pg=PA71">Symmetries</a>, Springer undergraduate mathematics series, Springer, p. 71, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781852332709" title="Special:BookSources/9781852332709"><bdi>9781852332709</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetries&rft.series=Springer+undergraduate+mathematics+series&rft.pages=71&rft.pub=Springer&rft.date=2001&rft.isbn=9781852332709&rft.aulast=Johnson&rft.aufirst=D.+L.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBecLeCWOjI4C%26pg%3DPA71&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-algebra-20"><a href="#cite_ref-algebra_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSahaiBist2003" class="citation cs2">Sahai, Vivek; Bist, Vikas (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VsoyRX_nHLkC&pg=PA152">Algebra</a>, Alpha Science International Ltd., p. 152, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781842651575" title="Special:BookSources/9781842651575"><bdi>9781842651575</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=152&rft.pub=Alpha+Science+International+Ltd.&rft.date=2003&rft.isbn=9781842651575&rft.aulast=Sahai&rft.aufirst=Vivek&rft.au=Bist%2C+Vikas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVsoyRX_nHLkC%26pg%3DPA152&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-ama-21"><a href="#cite_ref-ama_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman2015" class="citation cs2"><a href="/wiki/Joseph_J._Rotman" title="Joseph J. Rotman">Rotman, Joseph J.</a> (2015), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RGzK_DOTijsC&pg=PA450">Advanced Modern Algebra</a>, American Mathematical Society, p. 450, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821884201" title="Special:BookSources/9780821884201"><bdi>9780821884201</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Modern+Algebra&rft.pages=450&rft.pub=American+Mathematical+Society&rft.date=2015&rft.isbn=9780821884201&rft.aulast=Rotman&rft.aufirst=Joseph+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRGzK_DOTijsC%26pg%3DPA450&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-corner-22"><a href="#cite_ref-corner_22-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCorner2008" class="citation cs2">Corner, A. L. S. (2008), "Groups of units of orders in Q-algebras", Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 9–61, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110203035.9">10.1515/9783110203035.9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-019437-1" title="Special:BookSources/978-3-11-019437-1"><bdi>978-3-11-019437-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2513226">2513226</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Groups+of+units+of+orders+in+Q-algebras&rft.btitle=Models%2C+modules+and+abelian+groups&rft.pages=9-61&rft.pub=Walter+de+Gruyter%2C+Berlin&rft.date=2008&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2513226%23id-name%3DMR&rft_id=info%3Adoi%2F10.1515%2F9783110203035.9&rft.isbn=978-3-11-019437-1&rft.aulast=Corner&rft.aufirst=A.+L.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988">. See in particular the proof of Lemma H.4, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=khekRRwz0x0C&pg=PA36">p. 36</a>, which uses this fact. </li> <li id="cite_note-hatcher-23"><a href="#cite_ref-hatcher_23-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHatcher2002" class="citation cs2"><a href="/wiki/Allen_Hatcher" title="Allen Hatcher">Hatcher, Allen</a> (2002), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BjKs86kosqgC&pg=PA196">Algebraic Topology</a>, Cambridge University Press, p. 196, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521795401" title="Special:BookSources/9780521795401"><bdi>9780521795401</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Topology&rft.pages=196&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=9780521795401&rft.aulast=Hatcher&rft.aufirst=Allen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBjKs86kosqgC%26pg%3DPA196&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-vermani-24"><a href="#cite_ref-vermani_24-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVermani2004" class="citation cs2">Vermani, L. R. (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P27AtdajYRgC&pg=PA80">An Elementary Approach to Homological Algebra</a>, Monographs and Surveys in Pure and Applied Mathematics, CRC Press, p. 80, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780203484081" title="Special:BookSources/9780203484081"><bdi>9780203484081</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Elementary+Approach+to+Homological+Algebra&rft.series=Monographs+and+Surveys+in+Pure+and+Applied+Mathematics&rft.pages=80&rft.pub=CRC+Press&rft.date=2004&rft.isbn=9780203484081&rft.aulast=Vermani&rft.aufirst=L.+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP27AtdajYRgC%26pg%3DPA80&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-blass-25">^ <a href="#cite_ref-blass_25-0">a</a> <a href="#cite_ref-blass_25-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlass1979" class="citation cs2"><a href="/wiki/Andreas_Blass" title="Andreas Blass">Blass, Andreas</a> (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society, 255: 31–59, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1979-0542870-6">10.1090/S0002-9947-1979-0542870-6</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1998165">1998165</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0542870">0542870</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Injectivity%2C+projectivity%2C+and+the+axiom+of+choice&rft.volume=255&rft.pages=31-59&rft.date=1979&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D542870%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1998165%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1979-0542870-6&rft.aulast=Blass&rft.aufirst=Andreas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988">. For the connection to <a href="/wiki/Free_object" title="Free object">free objects</a>, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> in this model that is a subgroup of a free abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bigl (}\mathbb {Z} ^{(A)}{\bigr )}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\bigl (}\mathbb {Z} ^{(A)}{\bigr )}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46204fae45fb64b4fa3a8844a3e81a1be62bed64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.643ex; height:3.509ex;" alt="{\textstyle {\bigl (}\mathbb {Z} ^{(A)}{\bigr )}^{n}}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is a set of atoms and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is a finite integer. Blass writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free. </li> <li id="cite_note-FOOTNOTEHungerford1974Exercise&nbsp;4,_p.&nbsp;75-26"><a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;4,_p.&nbsp;75_26-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Exercise 4, p. 75. </li> <li id="cite_note-FOOTNOTEHungerford197470-27"><a href="#cite_ref-FOOTNOTEHungerford197470_27-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, p. 70. </li> <li id="cite_note-FOOTNOTEHungerford1974Theorem&nbsp;1.2,_p.&nbsp;73-28"><a href="#cite_ref-FOOTNOTEHungerford1974Theorem&nbsp;1.2,_p.&nbsp;73_28-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Theorem 1.2, p. 73. </li> <li id="cite_note-hm-29">^ <a href="#cite_ref-hm_29-0">a</a> <a href="#cite_ref-hm_29-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofmannMorris2006" class="citation cs2">Hofmann, Karl H.; Morris, Sidney A. (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YvcRi0x67mgC&pg=PA640">The Structure of Compact Groups: A Primer for Students - A Handbook for the Expert</a>, De Gruyter Studies in Mathematics, vol. 25 (2nd ed.), Walter de Gruyter, p. 640, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783110199772" title="Special:BookSources/9783110199772"><bdi>9783110199772</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Structure+of+Compact+Groups%3A+A+Primer+for+Students+-+A+Handbook+for+the+Expert&rft.series=De+Gruyter+Studies+in+Mathematics&rft.pages=640&rft.edition=2nd&rft.pub=Walter+de+Gruyter&rft.date=2006&rft.isbn=9783110199772&rft.aulast=Hofmann&rft.aufirst=Karl+H.&rft.au=Morris%2C+Sidney+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYvcRi0x67mgC%26pg%3DPA640&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-machi-30"><a href="#cite_ref-machi_30-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMachì2012" class="citation cs2">Machì, Antonio (2012), "Theorem 4.10", Groups: An introduction to ideas and methods of the theory of groups, Unitext, vol. 58, Milan: Springer, p. 172, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-88-470-2421-2">10.1007/978-88-470-2421-2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-88-470-2420-5" title="Special:BookSources/978-88-470-2420-5"><bdi>978-88-470-2420-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2987234">2987234</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theorem+4.10&rft.btitle=Groups%3A+An+introduction+to+ideas+and+methods+of+the+theory+of+groups&rft.place=Milan&rft.series=Unitext&rft.pages=172&rft.pub=Springer&rft.date=2012&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2987234%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-88-470-2421-2&rft.isbn=978-88-470-2420-5&rft.aulast=Mach%C3%AC&rft.aufirst=Antonio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-rotman-31"><a href="#cite_ref-rotman_31-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman1988" class="citation cs2"><a href="/wiki/Joseph_J._Rotman" title="Joseph J. Rotman">Rotman, Joseph J.</a> (1988), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=waq9mwUmcQgC&pg=PA61">An Introduction to Algebraic Topology</a>, Graduate Texts in Mathematics, vol. 119, Springer, pp. 61–62, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387966786" title="Special:BookSources/9780387966786"><bdi>9780387966786</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Algebraic+Topology&rft.series=Graduate+Texts+in+Mathematics&rft.pages=61-62&rft.pub=Springer&rft.date=1988&rft.isbn=9780387966786&rft.aulast=Rotman&rft.aufirst=Joseph+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dwaq9mwUmcQgC%26pg%3DPA61&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-johnson-32"><a href="#cite_ref-johnson_32-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson1980" class="citation cs2">Johnson, D. L. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, vol. 42, Cambridge University Press, p. 9, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-23108-4" title="Special:BookSources/978-0-521-23108-4"><bdi>978-0-521-23108-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0695161">0695161</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+the+Theory+of+Group+Presentations&rft.series=London+Mathematical+Society+lecture+note+series&rft.pages=9&rft.pub=Cambridge+University+Press&rft.date=1980&rft.isbn=978-0-521-23108-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0695161%23id-name%3DMR&rft.aulast=Johnson&rft.aufirst=D.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-lang-33"><a href="#cite_ref-lang_33-0">^</a> Appendix 2 §2, page 880 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2002" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (2002), Algebra, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 211 (Revised third ed.), New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95385-4" title="Special:BookSources/978-0-387-95385-4"><bdi>978-0-387-95385-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556">1878556</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0984.00001">0984.00001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=Revised+third&rft.pub=Springer-Verlag&rft.date=2002&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0984.00001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1878556%23id-name%3DMR&rft.isbn=978-0-387-95385-4&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHungerford1974Theorem&nbsp;1.6,_p.&nbsp;74-34"><a href="#cite_ref-FOOTNOTEHungerford1974Theorem&nbsp;1.6,_p.&nbsp;74_34-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Theorem 1.6, p. 74. </li> <li id="cite_note-FOOTNOTEJohnson200171–72-35"><a href="#cite_ref-FOOTNOTEJohnson200171–72_35-0">^</a> <a href="#CITEREFJohnson2001">Johnson (2001)</a>, pp. 71–72. </li> <li id="cite_note-norman-36"><a href="#cite_ref-norman_36-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNorman2012" class="citation cs2">Norman, Christopher (2012), "1.3 Uniqueness of the Smith Normal Form", Finitely Generated Abelian Groups and Similarity of Matrices over a Field, Springer undergraduate mathematics series, Springer, pp. 32–43, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012fgag.book.....N">2012fgag.book.....N</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781447127307" title="Special:BookSources/9781447127307"><bdi>9781447127307</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.3+Uniqueness+of+the+Smith+Normal+Form&rft.btitle=Finitely+Generated+Abelian+Groups+and+Similarity+of+Matrices+over+a+Field&rft.series=Springer+undergraduate+mathematics+series&rft.pages=32-43&rft.pub=Springer&rft.date=2012&rft_id=info%3Abibcode%2F2012fgag.book.....N&rft.isbn=9781447127307&rft.aulast=Norman&rft.aufirst=Christopher&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHungerford1974Exercise&nbsp;9,_p.&nbsp;75-37"><a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;9,_p.&nbsp;75_37-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Exercise 9, p. 75. </li> <li id="cite_note-FOOTNOTEHungerford1974Exercise&nbsp;10,_p.&nbsp;75-38"><a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;10,_p.&nbsp;75_38-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Exercise 10, p. 75. </li> <li id="cite_note-FOOTNOTEHungerford1974Exercise&nbsp;4,_p.&nbsp;198-39"><a href="#cite_ref-FOOTNOTEHungerford1974Exercise&nbsp;4,_p.&nbsp;198_39-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Exercise 4, p. 198. </li> <li id="cite_note-bridvog-40"><a href="#cite_ref-bridvog_40-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBridsonVogtmann2006" class="citation cs2"><a href="/wiki/Martin_Bridson" title="Martin Bridson">Bridson, Martin R.</a>; <a href="/wiki/Karen_Vogtmann" title="Karen Vogtmann">Vogtmann, Karen</a> (2006), "Automorphism groups of free groups, surface groups and free abelian groups", in <a href="/wiki/Benson_Farb" title="Benson Farb">Farb, Benson</a> (ed.), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74, Providence, Rhode Island: American Mathematical Society, pp. 301–316, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0507612">math/0507612</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fpspum%2F074%2F2264548">10.1090/pspum/074/2264548</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3838-9" title="Special:BookSources/978-0-8218-3838-9"><bdi>978-0-8218-3838-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2264548">2264548</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17710182">17710182</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Automorphism+groups+of+free+groups%2C+surface+groups+and+free+abelian+groups&rft.btitle=Problems+on+mapping+class+groups+and+related+topics&rft.place=Providence%2C+Rhode+Island&rft.series=Proceedings+of+Symposia+in+Pure+Mathematics&rft.pages=301-316&rft.pub=American+Mathematical+Society&rft.date=2006&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17710182%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2Fpspum%2F074%2F2264548&rft_id=info%3Aarxiv%2Fmath%2F0507612&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2264548%23id-name%3DMR&rft.isbn=978-0-8218-3838-9&rft.aulast=Bridson&rft.aufirst=Martin+R.&rft.au=Vogtmann%2C+Karen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-tolstykh-41"><a href="#cite_ref-tolstykh_41-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTolstykh2005" class="citation cs2">Tolstykh, Vladimir (2005), "What does the automorphism group of a free abelian group A know about A?", in <a href="/wiki/Andreas_Blass" title="Andreas Blass">Blass, Andreas</a>; Zhang, Yi (eds.), Logic and its Applications, Contemporary Mathematics, vol. 380, Providence, Rhode Island: American Mathematical Society, pp. 283–296, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0701752">math/0701752</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F380%2F07117">10.1090/conm/380/07117</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3474-9" title="Special:BookSources/978-0-8218-3474-9"><bdi>978-0-8218-3474-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2167584">2167584</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18107280">18107280</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=What+does+the+automorphism+group+of+a+free+abelian+group+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3EA%3C%2Fspan%3E+know+about+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3EA%3C%2Fspan%3E%3F&rft.btitle=Logic+and+its+Applications&rft.place=Providence%2C+Rhode+Island&rft.series=Contemporary+Mathematics&rft.pages=283-296&rft.pub=American+Mathematical+Society&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18107280%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2Fconm%2F380%2F07117&rft_id=info%3Aarxiv%2Fmath%2F0701752&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2167584%23id-name%3DMR&rft.isbn=978-0-8218-3474-9&rft.aulast=Tolstykh&rft.aufirst=Vladimir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHungerford1974Theorem&nbsp;1.4,_p.&nbsp;74-42"><a href="#cite_ref-FOOTNOTEHungerford1974Theorem&nbsp;1.4,_p.&nbsp;74_42-0">^</a> <a href="#CITEREFHungerford1974">Hungerford (1974)</a>, Theorem 1.4, p. 74. </li> <li id="cite_note-projective-choice-43"><a href="#cite_ref-projective-choice_43-0">^</a> The theorem that free abelian groups are projective is equivalent to the axiom of choice; see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoore2012" class="citation cs2">Moore, Gregory H. (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3RLGKcEjVIoC&pg=PR12">Zermelo's Axiom of Choice: Its Origins, Development, and Influence</a>, Courier Dover Publications, p. xii, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486488417" title="Special:BookSources/9780486488417"><bdi>9780486488417</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Zermelo%27s+Axiom+of+Choice%3A+Its+Origins%2C+Development%2C+and+Influence&rft.pages=xii&rft.pub=Courier+Dover+Publications&rft.date=2012&rft.isbn=9780486488417&rft.aulast=Moore&rft.aufirst=Gregory+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3RLGKcEjVIoC%26pg%3DPR12&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-griffith-44"><a href="#cite_ref-griffith_44-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffith1970" class="citation cs2"><a href="/wiki/Phillip_Griffith" title="Phillip Griffith">Griffith, Phillip A.</a> (1970), Infinite Abelian Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, p. 18, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-30870-7" title="Special:BookSources/0-226-30870-7"><bdi>0-226-30870-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinite+Abelian+Group+Theory&rft.series=Chicago+Lectures+in+Mathematics&rft.pages=18&rft.pub=University+of+Chicago+Press&rft.date=1970&rft.isbn=0-226-30870-7&rft.aulast=Griffith&rft.aufirst=Phillip+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-dctm-45"><a href="#cite_ref-dctm_45-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCavagnaroHaight2001" class="citation cs2"><a href="/wiki/Catherine_Cavagnaro" title="Catherine Cavagnaro">Cavagnaro, Catherine</a>; Haight, William T. II (2001), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ljvmahfSDtwC&pg=PA15">Dictionary of Classical and Theoretical Mathematics</a>, Comprehensive Dictionary of Mathematics, vol. 3, CRC Press, p. 15, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781584880509" title="Special:BookSources/9781584880509"><bdi>9781584880509</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionary+of+Classical+and+Theoretical+Mathematics&rft.series=Comprehensive+Dictionary+of+Mathematics&rft.pages=15&rft.pub=CRC+Press&rft.date=2001&rft.isbn=9781584880509&rft.aulast=Cavagnaro&rft.aufirst=Catherine&rft.au=Haight%2C+William+T.+II&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DljvmahfSDtwC%26pg%3DPA15&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-edelsbrunner-46"><a href="#cite_ref-edelsbrunner_46-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdelsbrunnerHarer2010" class="citation cs2"><a href="/wiki/Herbert_Edelsbrunner" title="Herbert Edelsbrunner">Edelsbrunner, Herbert</a>; Harer, John (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MDXa6gFRZuIC&pg=PA79">Computational Topology: An Introduction</a>, Providence, Rhode Island: American Mathematical Society, pp. 79–81, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821849255" title="Special:BookSources/9780821849255"><bdi>9780821849255</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Topology%3A+An+Introduction&rft.place=Providence%2C+Rhode+Island&rft.pages=79-81&rft.pub=American+Mathematical+Society&rft.date=2010&rft.isbn=9780821849255&rft.aulast=Edelsbrunner&rft.aufirst=Herbert&rft.au=Harer%2C+John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMDXa6gFRZuIC%26pg%3DPA79&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-dw-47">^ <a href="#cite_ref-dw_47-0">a</a> <a href="#cite_ref-dw_47-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDedekindWeber2012" class="citation cs2"><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind, Richard</a>; <a href="/wiki/Heinrich_Martin_Weber" title="Heinrich Martin Weber">Weber, Heinrich</a> (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Qxte2mhlEOYC&pg=PA13">Theory of Algebraic Functions of One Variable</a>, History of mathematics, vol. 39, Translated by <a href="/wiki/John_Stillwell" title="John Stillwell">John Stillwell</a>, American Mathematical Society, pp. 13–15, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821890349" title="Special:BookSources/9780821890349"><bdi>9780821890349</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Algebraic+Functions+of+One+Variable&rft.series=History+of+mathematics&rft.pages=13-15&rft.pub=American+Mathematical+Society&rft.date=2012&rft.isbn=9780821890349&rft.aulast=Dedekind&rft.aufirst=Richard&rft.au=Weber%2C+Heinrich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQxte2mhlEOYC%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-acrs-48"><a href="#cite_ref-acrs_48-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiranda1995" class="citation cs2">Miranda, Rick (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qjg6GOQaHNEC&pg=PA129">Algebraic Curves and Riemann Surfaces</a>, <a href="/wiki/Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>, vol. 5, American Mathematical Society, p. 129, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821802687" title="Special:BookSources/9780821802687"><bdi>9780821802687</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Curves+and+Riemann+Surfaces&rft.series=Graduate+Studies+in+Mathematics&rft.pages=129&rft.pub=American+Mathematical+Society&rft.date=1995&rft.isbn=9780821802687&rft.aulast=Miranda&rft.aufirst=Rick&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dqjg6GOQaHNEC%26pg%3DPA129&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-stein-szabo-49"><a href="#cite_ref-stein-szabo_49-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteinSzabó1994" class="citation cs2"><a href="/wiki/Sherman_K._Stein" title="Sherman K. Stein">Stein, Sherman K.</a>; Szabó, Sándor (1994), <a href="/wiki/Algebra_and_Tiling:_Homomorphisms_in_the_Service_of_Geometry" class="mw-redirect" title="Algebra and Tiling: Homomorphisms in the Service of Geometry">Algebra and Tiling: Homomorphisms in the Service of Geometry</a>, Carus Mathematical Monographs, vol. 25, Washington, DC: Mathematical Association of America, p. 198, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88385-028-1" title="Special:BookSources/0-88385-028-1"><bdi>0-88385-028-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1311249">1311249</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+and+Tiling%3A+Homomorphisms+in+the+Service+of+Geometry&rft.place=Washington%2C+DC&rft.series=Carus+Mathematical+Monographs&rft.pages=198&rft.pub=Mathematical+Association+of+America&rft.date=1994&rft.isbn=0-88385-028-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1311249%23id-name%3DMR&rft.aulast=Stein&rft.aufirst=Sherman+K.&rft.au=Szab%C3%B3%2C+S%C3%A1ndor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-higman-50"><a href="#cite_ref-higman_50-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHigman1940" class="citation cs2"><a href="/wiki/Graham_Higman" title="Graham Higman">Higman, Graham</a> (1940), "The units of group-rings", Proceedings of the London Mathematical Society, Second Series, 46: 231–248, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-46.1.231">10.1112/plms/s2-46.1.231</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0002137">0002137</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=The+units+of+group-rings&rft.volume=46&rft.pages=231-248&rft.date=1940&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-46.1.231&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0002137%23id-name%3DMR&rft.aulast=Higman&rft.aufirst=Graham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> <li id="cite_note-ayoub-ayoub-51"><a href="#cite_ref-ayoub-ayoub_51-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAyoubAyoub1969" class="citation cs2">Ayoub, Raymond G.; Ayoub, Christine (1969), "On the group ring of a finite abelian group", Bulletin of the Australian Mathematical Society, 1 (2): 245–261, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0004972700041496">10.1017/S0004972700041496</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0252526">0252526</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+Australian+Mathematical+Society&rft.atitle=On+the+group+ring+of+a+finite+abelian+group&rft.volume=1&rft.issue=2&rft.pages=245-261&rft.date=1969&rft_id=info%3Adoi%2F10.1017%2FS0004972700041496&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D252526%23id-name%3DMR&rft.aulast=Ayoub&rft.aufirst=Raymond+G.&rft.au=Ayoub%2C+Christine&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+abelian+group" class="Z3988"> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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Template:Short_description"," 10.72% 75.981 17 Template:Main_other"," 9.58% 67.909 13 Template:Sfnp"," 6.42% 45.470 56 Template:R/superscript"," 6.09% 43.137 1 Template:Good_article"]},"scribunto":{"limitreport-timeusage":{"value":"0.352","limit":"10.000"},"limitreport-memusage":{"value":7253800,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAyoubAyoub1969\"] = 1,\n [\"CITEREFBaer1937\"] = 1,\n [\"CITEREFBlass1979\"] = 1,\n [\"CITEREFBradley2005\"] = 1,\n [\"CITEREFBremner2011\"] = 1,\n [\"CITEREFBridsonVogtmann2006\"] = 1,\n [\"CITEREFCavagnaroHaight2001\"] = 1,\n [\"CITEREFCorner2008\"] = 1,\n [\"CITEREFDedekindWeber2012\"] = 1,\n [\"CITEREFEdelsbrunnerHarer2010\"] = 1,\n [\"CITEREFFuchs2015\"] = 1,\n [\"CITEREFGriffith1970\"] = 1,\n [\"CITEREFHartleyTurull1994\"] = 1,\n [\"CITEREFHatcher2002\"] = 1,\n [\"CITEREFHigman1940\"] = 1,\n [\"CITEREFHofmannMorris2006\"] = 1,\n [\"CITEREFHungerford1974\"] = 1,\n [\"CITEREFJohnson1980\"] = 1,\n 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1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-qmj8w","timestamp":"20241125133627","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Free abelian group","url":"https:\/\/en.wikipedia.org\/wiki\/Free_abelian_group","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1454111","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1454111","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-06-16T10:38:07Z","dateModified":"2024-10-30T10:35:38Z","headline":"commutative group whose elements are unique integer combinations of basis elements"}</script> </body> </html>