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class="vector-toc-numb">5</span> <span>Facts and theorems</span> </div> </a> <ul id="toc-Facts_and_theorems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_abelian_group" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Free_abelian_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Free abelian group</span> </div> </a> <ul id="toc-Free_abelian_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tarski's_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tarski's_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Tarski's problems</span> </div> </a> <ul id="toc-Tarski's_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Free group</span></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. 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href="https://ca.wikipedia.org/wiki/Grup_lliure" title="Grup lliure – Catalan" lang="ca" hreflang="ca" data-title="Grup lliure" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Freie_Gruppe" title="Freie Gruppe – German" lang="de" hreflang="de" data-title="Freie Gruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_libre" title="Grupo libre – Spanish" lang="es" hreflang="es" data-title="Grupo libre" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_libre" title="Groupe libre – French" lang="fr" hreflang="fr" data-title="Groupe libre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%9C%A0%EA%B5%B0" title="자유군 – Korean" lang="ko" hreflang="ko" data-title="자유군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_bebas" title="Grup bebas – Indonesian" lang="id" hreflang="id" data-title="Grup bebas" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_libero" title="Gruppo libero – Italian" lang="it" hreflang="it" data-title="Gruppo libero" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></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%97%D7%95%D7%A4%D7%A9%D7%99%D7%AA" title="חבורה חופשית – Hebrew" lang="he" hreflang="he" data-title="חבורה חופשית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szabad_csoport" title="Szabad csoport – Hungarian" lang="hu" hreflang="hu" data-title="Szabad csoport" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vrije_groep" title="Vrije groep – Dutch" lang="nl" hreflang="nl" data-title="Vrije groep" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%94%B1%E7%BE%A4" title="自由群 – Japanese" lang="ja" hreflang="ja" data-title="自由群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_wolna" title="Grupa wolna – Polish" lang="pl" hreflang="pl" data-title="Grupa wolna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_livre" title="Grupo livre – Portuguese" lang="pt" hreflang="pt" data-title="Grupo livre" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B2%D0%BE%D0%B1%D0%BE%D0%B4%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Свободная группа – Russian" lang="ru" hreflang="ru" data-title="Свободная группа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D1%96%D0%BB%D1%8C%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Вільна група – Ukrainian" lang="uk" hreflang="uk" data-title="Вільна група" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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.sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group"><i>p</i>-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</li> <li><a class="mw-selflink selflink">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:F2_Cayley_Graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/F2_Cayley_Graph.png/220px-F2_Cayley_Graph.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/F2_Cayley_Graph.png/330px-F2_Cayley_Graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/F2_Cayley_Graph.png/440px-F2_Cayley_Graph.png 2x" data-file-width="522" data-file-height="522" /></a><figcaption>Diagram showing the <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a> for the free group on two generators. Each vertex represents an element of the free group, and each edge represents multiplication by <i>a</i> or <i>b</i>.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>free group</b> <i>F</i><sub><i>S</i></sub> over a given set <i>S</i> consists of all <a href="/wiki/Word_(group_theory)" title="Word (group theory)">words</a> that can be built from members of <i>S</i>, considering two words to be different unless their equality follows from the <a href="/wiki/Group_(mathematics)#Definition" title="Group (mathematics)">group axioms</a> (e.g. <i>st</i> = <i>suu</i><sup>−1</sup><i>t</i> but <i>s</i> ≠ <i>t</i><sup>−1</sup> for <i>s</i>,<i>t</i>,<i>u</i> ∈ <i>S</i>). The members of <i>S</i> are called <b>generators</b> of <i>F</i><sub><i>S</i></sub>, and the number of generators is the <b>rank</b> of the free group. An arbitrary <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> is called <b>free</b> if it is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to <i>F</i><sub><i>S</i></sub> for some <a href="/wiki/Subset" title="Subset">subset</a> <i>S</i> of <i>G</i>, that is, if there is a subset <i>S</i> of <i>G</i> such that every element of <i>G</i> can be written in exactly one way as a product of finitely many elements of <i>S</i> and their inverses (disregarding trivial variations such as <i>st</i> = <i>suu</i><sup>−1</sup><i>t</i>). </p><p>A related but different notion is a <a href="/wiki/Free_abelian_group" title="Free abelian group">free abelian group</a>; both notions are particular instances of a <a href="/wiki/Free_object" title="Free object">free object</a> from <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>. As such, free groups are defined by their <a class="mw-selflink-fragment" href="#Universal_property">universal property</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Free groups first arose in the study of <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, as examples of <a href="/wiki/Fuchsian_group" title="Fuchsian group">Fuchsian groups</a> (discrete groups acting by <a href="/wiki/Isometry" title="Isometry">isometries</a> on the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic plane</a>). In an 1882 paper, <a href="/wiki/Walther_von_Dyck" title="Walther von Dyck">Walther von Dyck</a> pointed out that these groups have the simplest possible <a href="/wiki/Group_presentation" class="mw-redirect" title="Group presentation">presentations</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The algebraic study of free groups was initiated by <a href="/wiki/Jakob_Nielsen_(mathematician)" title="Jakob Nielsen (mathematician)">Jakob Nielsen</a> in 1924, who gave them their name and established many of their basic properties.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Max_Dehn" title="Max Dehn">Max Dehn</a> realized the connection with topology, and obtained the first proof of the full <a href="/wiki/Nielsen%E2%80%93Schreier_theorem" title="Nielsen–Schreier theorem">Nielsen–Schreier theorem</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Otto_Schreier" title="Otto Schreier">Otto Schreier</a> published an algebraic proof of this result in 1927,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Kurt_Reidemeister" title="Kurt Reidemeister">Kurt Reidemeister</a> included a comprehensive treatment of free groups in his 1932 book on <a href="/wiki/Combinatorial_topology" title="Combinatorial topology">combinatorial topology</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Later on in the 1930s, <a href="/wiki/Wilhelm_Magnus" title="Wilhelm Magnus">Wilhelm Magnus</a> discovered the connection between the <a href="/wiki/Lower_central_series" class="mw-redirect" title="Lower central series">lower central series</a> of free groups and <a href="/wiki/Free_Lie_algebra" title="Free Lie algebra">free Lie algebras</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The group (<b>Z</b>,+) of <a href="/wiki/Integer" title="Integer">integers</a> is free of rank 1; a generating set is <i>S</i> = {1}. The integers are also a <a href="/wiki/Free_abelian_group" title="Free abelian group">free abelian group</a>, although all free groups of rank <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a54ba6d06eab355fe1fb7e66f3a073de6db584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.616ex; height:2.343ex;" alt="{\displaystyle \geq 2}"></span> are non-abelian. A free group on a two-element set <i>S</i> occurs in the proof of the <a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a> and is described there. </p><p>On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order. </p><p>In <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of a <a href="/wiki/Bouquet_of_circles" class="mw-redirect" title="Bouquet of circles">bouquet of <i>k</i> circles</a> (a set of <i>k</i> loops having only one point in common) is the free group on a set of <i>k</i> elements. </p> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=3" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>free group</b> <i>F<sub>S</sub></i> with <b>free generating set</b> <i>S</i> can be constructed as follows. <i>S</i> is a set of symbols, and we suppose for every <i>s</i> in <i>S</i> there is a corresponding "inverse" symbol, <i>s</i><sup>−1</sup>, in a set <i>S</i><sup>−1</sup>. Let <i>T</i> = <i>S</i> ∪ <i>S</i><sup>−1</sup>, and define a <b><a href="/wiki/Word_(group_theory)" title="Word (group theory)">word</a></b> in <i>S</i> to be any written product of elements of <i>T</i>. That is, a word in <i>S</i> is an element of the <a href="/wiki/Monoid" title="Monoid">monoid</a> generated by <i>T</i>. The empty word is the word with no symbols at all. For example, if <i>S</i> = {<i>a</i>, <i>b</i>, <i>c</i>}, then <i>T</i> = {<i>a</i>, <i>a</i><sup>−1</sup>, <i>b</i>, <i>b</i><sup>−1</sup>, <i>c</i>, <i>c</i><sup>−1</sup>}, and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab^{3}c^{-1}ca^{-1}c\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>c</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>c</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab^{3}c^{-1}ca^{-1}c\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26151409883c8c4dfea972cd5adada055fcae1cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.585ex; height:2.676ex;" alt="{\displaystyle ab^{3}c^{-1}ca^{-1}c\,}"></span></dd></dl> <p>is a word in <i>S</i>. </p><p>If an element of <i>S</i> lies immediately next to its inverse, the word may be simplified by omitting the c, c<sup>−1</sup> pair: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab^{3}c^{-1}ca^{-1}c\;\;\longrightarrow \;\;ab^{3}\,a^{-1}c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>c</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>c</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟶</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab^{3}c^{-1}ca^{-1}c\;\;\longrightarrow \;\;ab^{3}\,a^{-1}c.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb7580343291185d465acfebf78dae9efd12659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:28.759ex; height:2.676ex;" alt="{\displaystyle ab^{3}c^{-1}ca^{-1}c\;\;\longrightarrow \;\;ab^{3}\,a^{-1}c.}"></span></dd></dl> <p>A word that cannot be simplified further is called <b>reduced</b>. </p><p>The free group <i>F<sub>S</sub></i> is defined to be the group of all reduced words in <i>S</i>, with <a href="/wiki/Concatenation" title="Concatenation">concatenation</a> of words (followed by reduction if necessary) as group operation. The identity is the empty word. </p><p>A reduced word is called <b>cyclically reduced</b> if its first and last letter are not inverse to each other. Every word is <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugate</a> to a cyclically reduced word, and a cyclically reduced conjugate of a cyclically reduced word is a cyclic permutation of the letters in the word. For instance <i>b</i><sup>−1</sup><i>abcb</i> is not cyclically reduced, but is conjugate to <i>abc</i>, which is cyclically reduced. The only cyclically reduced conjugates of <i>abc</i> are <i>abc</i>, <i>bca</i>, and <i>cab</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Universal_property">Universal property</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=4" title="Edit section: Universal property"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The free group <i>F<sub>S</sub></i> is the <a href="/wiki/Universal_(mathematics)" class="mw-redirect" title="Universal (mathematics)">universal</a> group generated by the set <i>S</i>. This can be formalized by the following <a href="/wiki/Universal_property" title="Universal property">universal property</a>: given any function <span class="texhtml mvar" style="font-style:italic;">f</span> from <i>S</i> to a group <i>G</i>, there exists a unique <a href="/wiki/Group_homomorphism" title="Group homomorphism">homomorphism</a> <i>φ</i>: <i>F<sub>S</sub></i> → <i>G</i> making the following <a href="/wiki/Commutative_diagram" title="Commutative diagram">diagram</a> commute (where the unnamed mapping denotes the <a href="/wiki/Inclusion_map" title="Inclusion map">inclusion</a> from <i>S</i> into <i>F<sub>S</sub></i>): </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Free_Group_Universal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Free_Group_Universal.svg/100px-Free_Group_Universal.svg.png" decoding="async" width="100" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Free_Group_Universal.svg/150px-Free_Group_Universal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Free_Group_Universal.svg/200px-Free_Group_Universal.svg.png 2x" data-file-width="277" data-file-height="261" /></a><figcaption></figcaption></figure> <p>That is, homomorphisms <i>F<sub>S</sub></i> → <i>G</i> are in one-to-one correspondence with functions <i>S</i> → <i>G</i>. For a non-free group, the presence of <a href="/wiki/Group_presentation" class="mw-redirect" title="Group presentation">relations</a> would restrict the possible images of the generators under a homomorphism. </p><p>To see how this relates to the constructive definition, think of the mapping from <i>S</i> to <i>F<sub>S</sub></i> as sending each symbol to a word consisting of that symbol. To construct <i>φ</i> for the given <span class="texhtml mvar" style="font-style:italic;">f</span>, first note that <i>φ</i> sends the empty word to the identity of <i>G</i> and it has to agree with <span class="texhtml mvar" style="font-style:italic;">f</span> on the elements of <i>S</i>. For the remaining words (consisting of more than one symbol), <i>φ</i> can be uniquely extended, since it is a homomorphism, i.e., <i>φ</i>(<i>ab</i>) = <i>φ</i>(<i>a</i>) <i>φ</i>(<i>b</i>). </p><p>The above property characterizes free groups up to <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>, and is sometimes used as an alternative definition. It is known as the <a href="/wiki/Universal_property" title="Universal property">universal property</a> of free groups, and the generating set <i>S</i> is called a <b>basis</b> for <i>F<sub>S</sub></i>. The basis for a free group is not uniquely determined. </p><p>Being characterized by a universal property is the standard feature of <a href="/wiki/Free_object" title="Free object">free objects</a> in <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>. In the language of <a href="/wiki/Category_theory" title="Category theory">category theory</a>, the construction of the free group (similar to most constructions of free objects) is a <a href="/wiki/Functor" title="Functor">functor</a> from the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a> to the <a href="/wiki/Category_of_groups" title="Category of groups">category of groups</a>. This functor is <a href="/wiki/Left_adjoint" class="mw-redirect" title="Left adjoint">left adjoint</a> to the <a href="/wiki/Forgetful_functor" title="Forgetful functor">forgetful functor</a> from groups to sets. </p> <div class="mw-heading mw-heading2"><h2 id="Facts_and_theorems">Facts and theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=5" title="Edit section: Facts and theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some properties of free groups follow readily from the definition: </p> <ol><li>Any group <i>G</i> is the homomorphic image of some free group <i>F<sub>S</sub></i>. Let <i>S</i> be a set of <i><a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generators</a></i> of <i>G</i>. The natural map <i>φ</i>: <i>F<sub>S</sub></i> → <i>G</i> is an <a href="/wiki/Epimorphism" title="Epimorphism">epimorphism</a>, which proves the claim. Equivalently, <i>G</i> is isomorphic to a <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> of some free group <i>F<sub>S</sub></i>. If <i>S</i> can be chosen to be finite here, then <i>G</i> is called <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a>. The kernel Ker(<i>φ)</i> is the set of all <i>relations</i> in the <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> of <i>G</i>; if Ker(<i>φ)</i> can be generated by the conjugates of finitely many elements of <i>F</i>, then <i>G</i> is finitely presented.</li> <li>If <i>S</i> has more than one element, then <i>F<sub>S</sub></i> is not <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>, and in fact the <a href="/wiki/Center_of_a_group" class="mw-redirect" title="Center of a group">center</a> of <i>F<sub>S</sub></i> is trivial (that is, consists only of the identity element).</li> <li>Two free groups <i>F<sub>S</sub></i> and <i>F<sub>T</sub></i> are isomorphic if and only if <i>S</i> and <i>T</i> have the same <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>. This cardinality is called the <b>rank</b> of the free group <i>F</i>. Thus for every <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a> <i>k</i>, there is, <a href="/wiki/Up_to" title="Up to">up to</a> isomorphism, exactly one free group of rank <i>k</i>.</li> <li>A free group of finite rank <i>n</i> > 1 has an <a href="/wiki/Exponential_growth" title="Exponential growth">exponential</a> <a href="/wiki/Growth_rate_(group_theory)" title="Growth rate (group theory)">growth rate</a> of order 2<i>n</i> − 1.</li></ol> <p>A few other related results are: </p> <ol><li>The <a href="/wiki/Nielsen%E2%80%93Schreier_theorem" title="Nielsen–Schreier theorem">Nielsen–Schreier theorem</a>: Every <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of a free group is free. Furthermore, if the free group <i>F</i> has rank <i>n</i> and the subgroup <i>H</i> has <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> <i>e</i> in <i>F</i>, then <i>H</i> is free of rank 1 + <i>e</i>(<i>n–</i>1).</li> <li>A free group of rank <i>k</i> clearly has subgroups of every rank less than <i>k</i>. Less obviously, a (<i>nonabelian!</i>) free group of rank at least 2 has subgroups of all <a href="/wiki/Countable_set" title="Countable set">countable</a> ranks.</li> <li>The <a href="/wiki/Commutator_subgroup" title="Commutator subgroup">commutator subgroup</a> of a free group of rank <i>k</i> > 1 has infinite rank; for example for F(<i>a</i>,<i>b</i>), it is freely generated by the <a href="/wiki/Commutator" title="Commutator">commutators</a> [<i>a</i><sup><i>m</i></sup>, <i>b</i><sup><i>n</i></sup>] for non-zero <i>m</i> and <i>n</i>.</li> <li>The free group in two elements is <a href="/wiki/SQ_universal" class="mw-redirect" title="SQ universal">SQ universal</a>; the above follows as any SQ universal group has subgroups of all countable ranks.</li> <li>Any group that <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a> on a tree, <a href="/wiki/Free_action" class="mw-redirect" title="Free action">freely</a> and preserving the <a href="/wiki/Oriented_graph" class="mw-redirect" title="Oriented graph">orientation</a>, is a free group of countable rank (given by 1 plus the <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of the <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">quotient</a> <a href="/wiki/Graph_theory" title="Graph theory">graph</a>).</li> <li>The <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a> of a free group of finite rank, with respect to a free generating set, is a <a href="/wiki/Tree_(graph_theory)" title="Tree (graph theory)">tree</a> on which the group acts freely, preserving the orientation. As a topological space (a one-dimensional <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a>), this Cayley graph Γ(<i>F</i>) is <a href="/wiki/Contractible_space" title="Contractible space">contractible</a>. For a finitely presented group <i>G,</i> the natural homomorphism defined above, <i>φ</i> : <i>F</i> → <i>G</i>, defines a <a href="/wiki/Covering_space" title="Covering space">covering map</a> of Cayley graphs <i>φ*</i> : Γ(<i>F</i>) → Γ(<i>G</i>), in fact a universal covering. Hence, the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of the Cayley graph Γ(<i>G</i>) is isomorphic to the kernel of <i>φ</i>, the normal subgroup of relations among the generators of <i>G</i>. The extreme case is when <i>G</i> = {<i>e</i>}, the trivial group, considered with as many generators as <i>F</i>, all of them trivial; the Cayley graph Γ(<i>G</i>) is a bouquet of circles, and its fundamental group is <i>F</i> itself.</li> <li>Any subgroup of a free group, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subset F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊂</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\subset F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/360b3243f2b8600f9e053743cb02d740c3a9fa36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.903ex; height:2.176ex;" alt="{\displaystyle H\subset F}"></span>, corresponds to a covering space of the bouquet of circles, namely to the <a href="/wiki/Schreier_coset_graph" title="Schreier coset graph">Schreier coset graph</a> of <i>F</i>/<i>H</i>. This can be used to give a topological proof of the Nielsen-Schreier theorem above.</li> <li>The <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> approach to these results, given in the work by P.J. Higgins below, is related to the use of <a href="/wiki/Covering_space" title="Covering space">covering spaces</a> above. It allows more powerful results, for example on <a href="/wiki/Grushko%27s_theorem" class="mw-redirect" title="Grushko's theorem">Grushko's theorem</a>, and a normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph.</li> <li><a href="/wiki/Grushko%27s_theorem" class="mw-redirect" title="Grushko's theorem">Grushko's theorem</a> has the consequence that if a subset <i>B</i> of a free group <i>F</i> on <i>n</i> elements generates <i>F</i> and has <i>n</i> elements, then <i>B</i> generates <i>F</i> freely.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Free_abelian_group">Free abelian group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=6" title="Edit section: Free abelian group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Further information: <a href="/wiki/Free_abelian_group" title="Free abelian group">Free abelian group</a></div> <p>The free abelian group on a set <i>S</i> is defined via its universal property in the analogous way, with obvious modifications: Consider a pair (<i>F</i>, <i>φ</i>), where <i>F</i> is an abelian group and <i>φ</i>: <i>S</i> → <i>F</i> is a function. <i>F</i> is said to be the <b>free abelian group on <i>S</i> with respect to <i>φ</i> </b> if for any abelian group <i>G</i> and any function <i>ψ</i>: <i>S</i> → <i>G</i>, there exists a unique homomorphism <i>f</i>: <i>F</i> → <i>G</i> such that </p> <dl><dd><i>f</i>(<i>φ</i>(<i>s</i>)) = <i>ψ</i>(<i>s</i>), for all <i>s</i> in <i>S</i>.</dd></dl> <p>The free abelian group on <i>S</i> can be explicitly identified as the free group F(<i>S</i>) modulo the subgroup generated by its commutators, [F(<i>S</i>), F(<i>S</i>)], i.e. its <a href="/wiki/Abelianisation" class="mw-redirect" title="Abelianisation">abelianisation</a>. In other words, the free abelian group on <i>S</i> is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as the rank of its abelianisation as a free abelian group. </p> <div class="mw-heading mw-heading2"><h2 id="Tarski's_problems"><span id="Tarski.27s_problems"></span>Tarski's problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=7" title="Edit section: Tarski's problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Around 1945, <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> asked whether the free groups on two or more generators have the same <a href="/wiki/Model_theory" title="Model theory">first-order theory</a>, and whether this theory is <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>. <a href="#CITEREFSela2006">Sela (2006)</a> answered the first question by showing that any two nonabelian free groups have the same first-order theory, and <a href="#CITEREFKharlampovichMyasnikov2006">Kharlampovich & Myasnikov (2006)</a> answered both questions, showing that this theory is decidable. </p><p>A similar unsolved (as of 2011) question in <a href="/wiki/Free_probability_theory" class="mw-redirect" title="Free probability theory">free probability theory</a> asks whether the <a href="/wiki/Von_Neumann_group_algebra" class="mw-redirect" title="Von Neumann group algebra">von Neumann group algebras</a> of any two non-abelian finitely generated free groups are isomorphic. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">Generating set of a group</a></li> <li><a href="/wiki/Presentation_of_a_group" title="Presentation of a group">Presentation of a group</a></li> <li><a href="/wiki/Nielsen_transformation" title="Nielsen transformation">Nielsen transformation</a>, a factorization of elements of the <a href="/wiki/Automorphism_group_of_a_free_group" title="Automorphism group of a free group">automorphism group of a free group</a></li> <li><a href="/wiki/Normal_form_for_free_groups_and_free_product_of_groups" title="Normal form for free groups and free product of groups">Normal form for free groups and free product of groups</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFvon_Dyck1882" class="citation journal cs1"><a href="/wiki/Walther_von_Dyck" title="Walther von Dyck">von Dyck, Walther</a> (1882). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304201754/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002246724&L=1">"Gruppentheoretische Studien (Group-theoretical Studies)"</a>. <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>. <b>20</b> (1): 1–44. <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%2FBF01443322">10.1007/BF01443322</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:179178038">179178038</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002246724&L=1">the original</a> on 2016-03-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Gruppentheoretische+Studien+%28Group-theoretical+Studies%29&rft.volume=20&rft.issue=1&rft.pages=1-44&rft.date=1882&rft_id=info%3Adoi%2F10.1007%2FBF01443322&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A179178038%23id-name%3DS2CID&rft.aulast=von+Dyck&rft.aufirst=Walther&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002246724%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsen1917" class="citation journal cs1"><a href="/wiki/Jakob_Nielsen_(mathematician)" title="Jakob Nielsen (mathematician)">Nielsen, Jakob</a> (1917). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160305141749/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266873&L=1">"Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden"</a>. <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>. <b>78</b> (1): 385–397. <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%2FBF01457113">10.1007/BF01457113</a>. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:46.0175.01">46.0175.01</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=1511907">1511907</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:119726936">119726936</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266873&L=1">the original</a> on 2016-03-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Die+Isomorphismen+der+allgemeinen+unendlichen+Gruppe+mit+zwei+Erzeugenden&rft.volume=78&rft.issue=1&rft.pages=385-397&rft.date=1917&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119726936%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1511907%23id-name%3DMR&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A46.0175.01%23id-name%3DJFM&rft_id=info%3Adoi%2F10.1007%2FBF01457113&rft.aulast=Nielsen&rft.aufirst=Jakob&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002266873%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span> </span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsen1921" class="citation journal cs1 cs1-prop-long-vol"><a href="/wiki/Jakob_Nielsen_(mathematician)" title="Jakob Nielsen (mathematician)">Nielsen, Jakob</a> (1921). "On calculation with noncommutative factors and its application to group theory. (Translated from Danish)". <i>The Mathematical Scientist</i>. 6 (1981) (2): 73–85.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Scientist&rft.atitle=On+calculation+with+noncommutative+factors+and+its+application+to+group+theory.+%28Translated+from+Danish%29&rft.volume=6+%281981%29&rft.issue=2&rft.pages=73-85&rft.date=1921&rft.aulast=Nielsen&rft.aufirst=Jakob&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsen1924" class="citation journal cs1"><a href="/wiki/Jakob_Nielsen_(mathematician)" title="Jakob Nielsen (mathematician)">Nielsen, Jakob</a> (1924). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160305073827/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002269813&L=1">"Die Isomorphismengruppe der freien Gruppen"</a>. <i>Mathematische Annalen</i>. <b>91</b> (3): 169–209. <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%2FBF01556078">10.1007/BF01556078</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:122577302">122577302</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002269813&L=1">the original</a> on 2016-03-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Die+Isomorphismengruppe+der+freien+Gruppen&rft.volume=91&rft.issue=3&rft.pages=169-209&rft.date=1924&rft_id=info%3Adoi%2F10.1007%2FBF01556078&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122577302%23id-name%3DS2CID&rft.aulast=Nielsen&rft.aufirst=Jakob&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002269813%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMagnusMoufang1954" class="citation journal cs1"><a href="/wiki/Wilhelm_Magnus" title="Wilhelm Magnus">Magnus, Wilhelm</a>; <a href="/wiki/Ruth_Moufang" title="Ruth Moufang">Moufang, Ruth</a> (1954). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160305072926/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002283808&L=1">"Max Dehn zum Gedächtnis"</a>. <i>Mathematische Annalen</i>. <b>127</b> (1): 215–227. <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%2FBF01361121">10.1007/BF01361121</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:119917209">119917209</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002283808&L=1">the original</a> on 2016-03-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Max+Dehn+zum+Ged%C3%A4chtnis&rft.volume=127&rft.issue=1&rft.pages=215-227&rft.date=1954&rft_id=info%3Adoi%2F10.1007%2FBF01361121&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119917209%23id-name%3DS2CID&rft.aulast=Magnus&rft.aufirst=Wilhelm&rft.au=Moufang%2C+Ruth&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002283808%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchreier1928" class="citation journal cs1"><a href="/wiki/Otto_Schreier" title="Otto Schreier">Schreier, Otto</a> (1928). "Die Untergruppen der freien Gruppen". <i>Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg</i>. <b>5</b>: 161–183. <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%2FBF02952517">10.1007/BF02952517</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:121888949">121888949</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Abhandlungen+aus+dem+Mathematischen+Seminar+der+Universit%C3%A4t+Hamburg&rft.atitle=Die+Untergruppen+der+freien+Gruppen&rft.volume=5&rft.pages=161-183&rft.date=1928&rft_id=info%3Adoi%2F10.1007%2FBF02952517&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121888949%23id-name%3DS2CID&rft.aulast=Schreier&rft.aufirst=Otto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReidemeister1972" class="citation book cs1"><a href="/wiki/Kurt_Reidemeister" title="Kurt Reidemeister">Reidemeister, Kurt</a> (1972) [1932]. <i>Einführung in die kombinatorische Topologie</i>. Darmstadt: Wissenschaftliche Buchgesellschaft.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einf%C3%BChrung+in+die+kombinatorische+Topologie&rft.place=Darmstadt&rft.pub=Wissenschaftliche+Buchgesellschaft&rft.date=1972&rft.aulast=Reidemeister&rft.aufirst=Kurt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Free_group&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKharlampovichMyasnikov2006" class="citation journal cs1">Kharlampovich, Olga; Myasnikov, Alexei (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jalgebra.2006.03.033">"Elementary theory of free non-abelian groups"</a>. <i><a href="/wiki/Journal_of_Algebra" title="Journal of Algebra">Journal of Algebra</a></i>. <b>302</b> (2): 451–552. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jalgebra.2006.03.033">10.1016/j.jalgebra.2006.03.033</a></span>. <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=2293770">2293770</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Algebra&rft.atitle=Elementary+theory+of+free+non-abelian+groups&rft.volume=302&rft.issue=2&rft.pages=451-552&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.jalgebra.2006.03.033&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2293770%23id-name%3DMR&rft.aulast=Kharlampovich&rft.aufirst=Olga&rft.au=Myasnikov%2C+Alexei&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.jalgebra.2006.03.033&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></li> <li>W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory", Dover (1976).</li> <li>P.J. Higgins, 1971, "Categories and Groupoids", van Nostrand, {New York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSela2006" class="citation journal cs1"><a href="/wiki/Zlil_Sela" title="Zlil Sela">Sela, Zlil</a> (2006). "Diophantine geometry over groups. VI. The elementary theory of a free group". <i>Geom. Funct. Anal</i>. <b>16</b> (3): 707–730. <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%2Fs00039-006-0565-8">10.1007/s00039-006-0565-8</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=2238945">2238945</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:123197664">123197664</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geom.+Funct.+Anal.&rft.atitle=Diophantine+geometry+over+groups.+VI.+The+elementary+theory+of+a+free+group.&rft.volume=16&rft.issue=3&rft.pages=707-730&rft.date=2006&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2238945%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123197664%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00039-006-0565-8&rft.aulast=Sela&rft.aufirst=Zlil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span></li> <li><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a>, <i>Trees</i>, Springer (2003) (English translation of "arbres, amalgames, SL<sub>2</sub>", 3rd edition, <i>astérisque</i> <b>46</b> (1983))</li> <li>P.J. Higgins, <i><a rel="nofollow" class="external text" href="https://doi.org/10.1112/jlms/s2-13.1.145">The fundamental groupoid of a graph of groups</a></i>, <a href="/wiki/Journal_of_the_London_Mathematical_Society" class="mw-redirect" title="Journal of the London Mathematical Society">Journal of the London Mathematical Society</a> (2) <b>13</b> (1976), no. 1, 145–149.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAluffi2009" class="citation book cs1">Aluffi, Paolo (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=deWkZWYbyHQC&pg=PA70"><i>Algebra: Chapter 0</i></a>. AMS Bookstore. p. 70. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4781-7" title="Special:BookSources/978-0-8218-4781-7"><bdi>978-0-8218-4781-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra%3A+Chapter+0&rft.pages=70&rft.pub=AMS+Bookstore&rft.date=2009&rft.isbn=978-0-8218-4781-7&rft.aulast=Aluffi&rft.aufirst=Paolo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdeWkZWYbyHQC%26pg%3DPA70&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrillet2007" class="citation book cs1">Grillet, Pierre Antoine (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LJtyhu8-xYwC&pg=PA27"><i>Abstract algebra</i></a>. Springer. p. 27. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-71567-4" title="Special:BookSources/978-0-387-71567-4"><bdi>978-0-387-71567-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+algebra&rft.pages=27&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-71567-4&rft.aulast=Grillet&rft.aufirst=Pierre+Antoine&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLJtyhu8-xYwC%26pg%3DPA27&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFree+group" class="Z3988"></span>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Free_group&oldid=1225639942">https://en.wikipedia.org/w/index.php?title=Free_group&oldid=1225639942</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Geometric_group_theory" title="Category:Geometric group theory">Geometric group theory</a></li><li><a href="/wiki/Category:Combinatorial_group_theory" title="Category:Combinatorial group theory">Combinatorial group theory</a></li><li><a href="/wiki/Category:Free_algebraic_structures" title="Category:Free algebraic structures">Free algebraic structures</a></li><li><a href="/wiki/Category:Properties_of_groups" title="Category:Properties of groups">Properties of groups</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 25 May 2024, at 19:40<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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