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class="vector-toc-list"> <li id="toc-Normal_series,_subnormal_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_series,_subnormal_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Normal series, subnormal series</span> </div> </a> <ul id="toc-Normal_series,_subnormal_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Length" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Length"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Length</span> </div> </a> <ul id="toc-Length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ascending_series,_descending_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ascending_series,_descending_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Ascending series, descending series</span> </div> </a> <ul id="toc-Ascending_series,_descending_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noetherian_groups,_Artinian_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noetherian_groups,_Artinian_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Noetherian groups, Artinian groups</span> </div> </a> <ul id="toc-Noetherian_groups,_Artinian_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_and_transfinite_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_and_transfinite_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Infinite and transfinite series</span> </div> </a> <ul id="toc-Infinite_and_transfinite_series-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Comparison_of_series" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison_of_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Comparison of series</span> </div> </a> <ul id="toc-Comparison_of_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Maximal_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximal_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Maximal series</span> </div> </a> <ul id="toc-Maximal_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solvable_and_nilpotent" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solvable_and_nilpotent"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Solvable and nilpotent</span> </div> </a> <ul id="toc-Solvable_and_nilpotent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functional_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functional_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Functional series</span> </div> </a> <ul id="toc-Functional_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-p-series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#p-series"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span><i>p</i>-series</span> 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data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Normal_series&redirect=no" class="mw-redirect" title="Normal series">Normal series</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, specifically <a href="/wiki/Group_theory" title="Group theory">group theory</a>, a <b>subgroup series</b> of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</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 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></span><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}"></span> is a <a href="/wiki/Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">chain</a> of <a href="/wiki/Subgroup" title="Subgroup">subgroups</a>: </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 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd48d68c7754016b342ff25a5ae13980262720c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.761ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G}"></span></dd></dl> <p>where <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 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></span><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}"></span> is the <a href="/wiki/Trivial_group" title="Trivial group">trivial subgroup</a>. Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and <a href="#Functional_series">several subgroup series</a> can be invariantly defined and are important invariants of groups. A subgroup series is used in the <a href="/wiki/Subgroup_method" title="Subgroup method">subgroup method</a>. </p><p>Subgroup series are a special example of the use of <a href="/wiki/Filtration_(mathematics)" title="Filtration (mathematics)">filtrations</a> in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Normal_series,_subnormal_series"><span id="Normal_series.2C_subnormal_series"></span>Normal series, subnormal series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=2" title="Edit section: Normal series, subnormal series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>subnormal series</b> (also <b>normal series</b>, <b>normal tower</b>, <b>subinvariant series</b>, or just <b>series</b>) of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> is a sequence of <a href="/wiki/Subgroup" title="Subgroup">subgroups</a>, each a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of the next one. In a standard notation </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 1=A_{0}\triangleleft A_{1}\triangleleft \cdots \triangleleft A_{n}=G.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>◃</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>◃</mo> <mo>⋯</mo> <mo>◃</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>G</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\triangleleft A_{1}\triangleleft \cdots \triangleleft A_{n}=G.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f80f030efa4e0cfb67a5a4375e5a0b0ad1c71f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.697ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\triangleleft A_{1}\triangleleft \cdots \triangleleft A_{n}=G.}"></span></dd></dl> <p>There is no requirement made that <i>A</i><sub><i>i</i></sub> be a normal subgroup of <i>G</i>, only a normal subgroup of <i>A</i><sub><i>i</i> +1</sub>. The <a href="/wiki/Quotient_group" title="Quotient group">quotient groups</a> <i>A</i><sub><i>i</i> +1</sub>/<i>A</i><sub><i>i</i></sub> are called the <b>factor groups</b> of the series. </p><p>If in addition each <i>A</i><sub><i>i</i></sub> is normal in <i>G</i>, then the series is called a <b>normal series</b>, when this term is not used for the weaker sense, or an <b>invariant series</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Length">Length</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=3" title="Edit section: Length"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A series with the additional property that <i>A</i><sub><i>i</i></sub> ≠ <i>A</i><sub><i>i</i> +1</sub> for all <i>i</i> is called a series <i>without repetition</i>; equivalently, each <i>A</i><sub><i>i</i></sub> is a proper subgroup of <i>A</i><sub><i>i</i> +1</sub>. The <i>length</i> of a series is the number of strict inclusions <i>A</i><sub><i>i</i></sub> < <i>A</i><sub><i>i</i> +1</sub>. If the series has no repetition then the length is <i>n</i>. </p><p>For a subnormal series, the length is the number of <a href="/wiki/Trivial_group" title="Trivial group">non-trivial</a> factor groups. Every nontrivial group has a normal series of length 1, namely <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 1\triangleleft G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>◃</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\triangleleft G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c517e353dd41328a584bec48b0d07eab1e72c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.184ex; height:2.176ex;" alt="{\displaystyle 1\triangleleft G}"></span>, and any nontrivial proper normal subgroup gives a normal series of length 2. For <a href="/wiki/Simple_group" title="Simple group">simple groups</a>, the trivial series of length 1 is the longest subnormal series possible. </p> <div class="mw-heading mw-heading3"><h3 id="Ascending_series,_descending_series"><span id="Ascending_series.2C_descending_series"></span>Ascending series, descending series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=4" title="Edit section: Ascending series, descending series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Series can be notated in either ascending order: </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 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd48d68c7754016b342ff25a5ae13980262720c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.761ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G}"></span></dd></dl> <p>or descending order: </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 G=B_{0}\geq B_{1}\geq \cdots \geq B_{n}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <mo>⋯</mo> <mo>≥</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=B_{0}\geq B_{1}\geq \cdots \geq B_{n}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122899a8ebebca8df5a8bdb0392c1beaa650a79f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.471ex; height:2.509ex;" alt="{\displaystyle G=B_{0}\geq B_{1}\geq \cdots \geq B_{n}=1.}"></span></dd></dl> <p>For a given finite series, there is no distinction between an "ascending series" or "descending series" beyond notation. For <i>infinite</i> series however, there is a distinction: the ascending series </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 1=A_{0}\leq A_{1}\leq \cdots \leq G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d999d6a344dba5bc6354fc07f3b3b8e6f8403168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.701ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq G}"></span></dd></dl> <p>has a smallest term, a second smallest term, and so forth, but no largest proper term, no second largest term, and so forth, while conversely the descending series </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 G=B_{0}\geq B_{1}\geq \cdots \geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <mo>⋯</mo> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=B_{0}\geq B_{1}\geq \cdots \geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498d779e5ffeea6513b3acdcfc99dd4c92977066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.743ex; height:2.509ex;" alt="{\displaystyle G=B_{0}\geq B_{1}\geq \cdots \geq 1}"></span></dd></dl> <p>has a largest term, but no smallest proper term. </p><p>Further, given a recursive formula for producing a series, the terms produced are either ascending or descending, and one calls the resulting series an ascending or descending series, respectively. For instance the <a href="/wiki/Derived_series" class="mw-redirect" title="Derived series">derived series</a> and <a href="/wiki/Lower_central_series" class="mw-redirect" title="Lower central series">lower central series</a> are descending series, while the <a href="/wiki/Upper_central_series" class="mw-redirect" title="Upper central series">upper central series</a> is an ascending series. </p> <div class="mw-heading mw-heading3"><h3 id="Noetherian_groups,_Artinian_groups"><span id="Noetherian_groups.2C_Artinian_groups"></span>Noetherian groups, Artinian groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=5" title="Edit section: Noetherian groups, Artinian groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A group that satisfies the <a href="/wiki/Ascending_chain_condition" title="Ascending chain condition">ascending chain condition</a> (ACC) on subgroups is called a <b>Noetherian group</b>, and a group that satisfies the <a href="/wiki/Descending_chain_condition" class="mw-redirect" title="Descending chain condition">descending chain condition</a> (DCC) is called an <b>Artinian group</b> (not to be confused with <a href="/wiki/Artin_group" class="mw-redirect" title="Artin group">Artin groups</a>), by analogy with <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian rings</a> and <a href="/wiki/Artinian_ring" title="Artinian ring">Artinian rings</a>. The ACC is equivalent to the <b>maximal condition</b>: every <a href="/wiki/Empty_set" title="Empty set">non-empty</a> collection of subgroups has a maximal member, and the DCC is equivalent to the analogous <b>minimal condition</b>. </p><p>A group can be Noetherian but not Artinian, such as the <a href="/wiki/Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a>, and unlike for <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, a group can be Artinian but not Noetherian, such as the <a href="/wiki/Pr%C3%BCfer_group" title="Prüfer group">Prüfer group</a>. Every finite group is clearly Noetherian and Artinian. </p><p><a href="/wiki/Group_homomorphism" title="Group homomorphism">Homomorphic</a> <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">images</a> and subgroups of Noetherian groups are Noetherian, and an <a href="/wiki/Group_extension" title="Group extension">extension</a> of a Noetherian group by a Noetherian group is Noetherian. Analogous results hold for Artinian groups. </p><p>Noetherian groups are equivalently those such that every subgroup is <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a>, which is stronger than the group itself being finitely generated: the <a href="/wiki/Free_group" title="Free group">free group</a> on 2 or finitely more generators is finitely generated, but contains free groups of infinite rank. </p><p>Noetherian groups need not be finite extensions of <a href="/wiki/Polycyclic_group" title="Polycyclic group">polycyclic groups</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> </p> <div class="mw-heading mw-heading3"><h3 id="Infinite_and_transfinite_series">Infinite and transfinite series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=6" title="Edit section: Infinite and transfinite series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Infinite subgroup series can also be defined and arise naturally, in which case the specific (<a href="/wiki/Total_order" title="Total order">totally ordered</a>) indexing set becomes important, and there is a distinction between ascending and descending series. An ascending series <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 1=A_{0}\leq A_{1}\leq \cdots \leq G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d999d6a344dba5bc6354fc07f3b3b8e6f8403168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.701ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq G}"></span> where the <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 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span> are indexed by the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> may simply be called an <b>infinite ascending series</b>, and conversely for an <b>infinite descending series</b>. If the subgroups are more generally <a href="/wiki/Ordinal_number#Indexing_classes_of_ordinals" title="Ordinal number">indexed by ordinal numbers</a>, one obtains a <b>transfinite series</b>,<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> such as this ascending series: </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 1=A_{0}\leq A_{1}\leq \cdots \leq A_{\omega }\leq A_{\omega +1}=G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{\omega }\leq A_{\omega +1}=G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce96dc347c5f4f09d07deeb93a36b510d959958c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.994ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{\omega }\leq A_{\omega +1}=G}"></span></dd></dl> <p>Given a recursive formula for producing a series, one can define a transfinite series by <a href="/wiki/Transfinite_recursion" class="mw-redirect" title="Transfinite recursion">transfinite recursion</a> by defining the series at <a href="/wiki/Limit_ordinal" title="Limit ordinal">limit ordinals</a> by <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 A_{\lambda }:=\bigcup _{\alpha <\lambda }A_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>:=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo><</mo> <mi>λ</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\lambda }:=\bigcup _{\alpha <\lambda }A_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6690f82cb6f7ceccc75a34caec4048f931d775cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.382ex; height:5.676ex;" alt="{\displaystyle A_{\lambda }:=\bigcup _{\alpha <\lambda }A_{\alpha }}"></span> (for ascending series) or <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 A_{\lambda }:=\bigcap _{\alpha <\lambda }A_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo>:=</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo><</mo> <mi>λ</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\lambda }:=\bigcap _{\alpha <\lambda }A_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92fd19c72cc852b56fbfbad152675911a27bc159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.382ex; height:5.676ex;" alt="{\displaystyle A_{\lambda }:=\bigcap _{\alpha <\lambda }A_{\alpha }}"></span> (for descending series). Fundamental examples of this construction are the transfinite <a href="/wiki/Lower_central_series" class="mw-redirect" title="Lower central series">lower central series</a> and <a href="/wiki/Upper_central_series" class="mw-redirect" title="Upper central series">upper central series</a>. </p><p>Other totally ordered sets arise rarely, if ever, as indexing sets of subgroup series.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2008)">citation needed</span></a></i>]</sup> For instance, one can define but rarely sees naturally occurring bi-infinite subgroup series (series indexed by the <a href="/wiki/Integer" title="Integer">integers</a>): </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 1\leq \cdots \leq A_{-1}\leq A_{0}\leq A_{1}\leq \cdots \leq G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq \cdots \leq A_{-1}\leq A_{0}\leq A_{1}\leq \cdots \leq G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07bc7fbaeb4429c086276f732511a558888a53e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.697ex; height:2.509ex;" alt="{\displaystyle 1\leq \cdots \leq A_{-1}\leq A_{0}\leq A_{1}\leq \cdots \leq G}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Comparison_of_series">Comparison of series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=7" title="Edit section: Comparison of series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>refinement</i> of a series is another series containing each of the terms of the original series. Two subnormal series are said to be <i>equivalent</i> or <i>isomorphic</i> if there is a <a href="/wiki/Bijection" title="Bijection">bijection</a> between the sets of their factor groups such that the corresponding factor groups are <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a>. Refinement gives a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> on series, up to equivalence, and they form a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a>, while subnormal series and normal series form sublattices. The existence of the supremum of two subnormal series is the <a href="/wiki/Schreier_refinement_theorem" title="Schreier refinement theorem">Schreier refinement theorem</a>. Of particular interest are <i>maximal</i> series without repetition. </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=Subgroup_series&action=edit&section=8" title="Edit section: Examples"><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 selfref">See also: <a href="/wiki/Category:Subgroup_series" title="Category:Subgroup series">Category:Subgroup series</a></div> <div class="mw-heading mw-heading3"><h3 id="Maximal_series">Maximal series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=9" title="Edit section: Maximal series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <b><a href="/wiki/Composition_series" title="Composition series">composition series</a></b> is a maximal <i>subnormal</i> series.</li></ul> <dl><dd>Equivalently, a subnormal series for which each of the <i>A</i><sub><i>i</i></sub> is a <a href="/wiki/Maximal_subgroup" title="Maximal subgroup">maximal</a> normal subgroup of <i>A</i><sub><i>i</i> +1</sub>. Equivalently, a composition series is a subnormal series for which each of the factor groups are <a href="/wiki/Simple_group" title="Simple group">simple</a>.</dd></dl> <ul><li>A <b><a href="/wiki/Chief_series" title="Chief series">chief series</a></b> is a maximal <i>normal</i> series.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Solvable_and_nilpotent">Solvable and nilpotent</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=10" title="Edit section: Solvable and nilpotent"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <b><a href="/wiki/Solvable_group" title="Solvable group">solvable group</a></b>, or soluble group, is one with a subnormal series whose factor groups are all <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>.</li> <li>A <b><a href="/wiki/Nilpotent_series" class="mw-redirect" title="Nilpotent series">nilpotent series</a></b> is a subnormal series such that successive quotients are <a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a>.</li></ul> <dl><dd>A nilpotent series exists if and only if the group is <a href="/wiki/Solvable_group" title="Solvable group">solvable</a>.</dd></dl> <ul><li>A <b><a href="/wiki/Central_series" title="Central series">central series</a></b> is a subnormal series such that successive quotients are <a href="/wiki/Center_(group)" class="mw-redirect" title="Center (group)">central</a>, i.e. given the above series, <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 A_{i+1}/A_{i}\subseteq Z(G/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> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊆</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i+1}/A_{i}\subseteq Z(G/A_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0961a78d8d1e68d66a60394260b9287bae6a45ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.469ex; height:2.843ex;" alt="{\displaystyle A_{i+1}/A_{i}\subseteq Z(G/A_{i})}"></span> for <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 i=0,1,\ldots ,n-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=0,1,\ldots ,n-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4120cb6112ac26d9c6d59f5965636abc4faf235c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.836ex; height:2.509ex;" alt="{\displaystyle i=0,1,\ldots ,n-2}"></span>.</li></ul> <dl><dd>A central series exists if and only if the group is <a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Functional_series">Functional series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=11" title="Edit section: Functional series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some subgroup series are defined <a href="/wiki/Category:Functional_subgroups" title="Category:Functional subgroups">functionally</a>, in terms of subgroups such as the center and operations such as the commutator. These include: </p> <ul><li><a href="/wiki/Lower_central_series" class="mw-redirect" title="Lower central series">Lower central series</a></li> <li><a href="/wiki/Upper_central_series" class="mw-redirect" title="Upper central series">Upper central series</a></li> <li><a href="/wiki/Derived_series" class="mw-redirect" title="Derived series">Derived series</a></li> <li><a href="/wiki/Lower_Fitting_series" class="mw-redirect" title="Lower Fitting series">Lower Fitting series</a></li> <li><a href="/wiki/Upper_Fitting_series" class="mw-redirect" title="Upper Fitting series">Upper Fitting series</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="p-series"><i>p</i>-series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup_series&action=edit&section=12" title="Edit section: p-series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are series coming from subgroups of prime power order or prime power index, related to ideas such as <a href="/wiki/Sylow_subgroup" class="mw-redirect" title="Sylow subgroup">Sylow subgroups</a>. </p> <ul><li><a href="/wiki/Lower_p-series" class="mw-redirect" title="Lower p-series">Lower <i>p</i>-series</a></li> <li><a href="/wiki/Upper_p-series" class="mw-redirect" title="Upper p-series">Upper <i>p</i>-series</a></li></ul> <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=Subgroup_series&action=edit&section=13" title="Edit section: References"><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="CITEREFOl'shanskii,_A._Yu.1979" class="citation journal cs1">Ol'shanskii, A. Yu. (1979). "Infinite Groups with Cyclic Subgroups". <i>Soviet Math. Dokl</i>. <b>20</b>: 343–346.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Math.+Dokl.&rft.atitle=Infinite+Groups+with+Cyclic+Subgroups&rft.volume=20&rft.pages=343-346&rft.date=1979&rft.au=Ol%27shanskii%2C+A.+Yu.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup+series" class="Z3988"></span> (English translation of <i>Dokl. Akad. Nauk SSSR</i>, <b>245</b>, 785–787)</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="CITEREFSharipov2009" class="citation arxiv cs1">Sharipov, R.A. (2009). "Transfinite normal and composition series of groups". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0908.2257">0908.2257</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.GR">math.GR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Transfinite+normal+and+composition+series+of+groups&rft.date=2009&rft_id=info%3Aarxiv%2F0908.2257&rft.aulast=Sharipov&rft.aufirst=R.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup+series" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" 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=Subgroup_series&oldid=1043865885#Normal_series.2C_subnormal_series">https://en.wikipedia.org/w/index.php?title=Subgroup_series&oldid=1043865885#Normal_series.2C_subnormal_series</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">Category</a>: <ul><li><a href="/wiki/Category:Subgroup_series" title="Category:Subgroup series">Subgroup series</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:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_January_2008" title="Category:Articles with unsourced statements from January 2008">Articles with unsourced statements from January 2008</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 12 September 2021, at 11:31<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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