SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Subgroup series</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Subgroup series</span></span>

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, specifically <a href="Group_theory" title="Group theory">group theory</a>, a <b>subgroup series</b> of a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" 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}" loading="lazy"></span> is a <a href="Chain_(order_theory)" class="mw-redirect" title="Chain (order theory)">chain</a> of <a href="Subgroup" title="Subgroup">subgroups</a>:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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="./3cd48d68c7754016b342ff25a5ae13980262720c.svg" 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}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" 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}" loading="lazy"></span> is the <a href="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="Subgroup_method" class="mw-redirect" title="Subgroup method">subgroup method</a>.

</p><p>Subgroup series are a special example of the use of <a href="Filtration_(mathematics)" title="Filtration (mathematics)">filtrations</a> in <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Normal_series,_subnormal_series">Normal series, subnormal series</h3></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="Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> is a sequence of <a href="Subgroup" title="Subgroup">subgroups</a>, each a <a href="Normal_subgroup" title="Normal subgroup">normal subgroup</a> of the next one. In a standard notation

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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="./2f80f030efa4e0cfb67a5a4375e5a0b0ad1c71f5.svg" 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.}" loading="lazy"></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="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></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="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 mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\triangleleft G}</annotation>

</semantics>

</math></span><img src="./07c517e353dd41328a584bec48b0d07eab1e72c4.svg" 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}" loading="lazy"></span>, and any nontrivial proper normal subgroup gives a normal series of length 2. For <a href="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">Ascending series, descending series</h3></div>

<p>Series can be notated in either ascending order:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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="./3cd48d68c7754016b342ff25a5ae13980262720c.svg" 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}" loading="lazy"></span></dd></dl>

<p>or descending order:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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="./122899a8ebebca8df5a8bdb0392c1beaa650a79f.svg" 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.}" loading="lazy"></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>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq G}</annotation>

</semantics>

</math></span><img src="./d999d6a344dba5bc6354fc07f3b3b8e6f8403168.svg" 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}" loading="lazy"></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>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=B_{0}\geq B_{1}\geq \cdots \geq 1}</annotation>

</semantics>

</math></span><img src="./498d779e5ffeea6513b3acdcfc99dd4c92977066.svg" 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}" loading="lazy"></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="Derived_series" class="mw-redirect" title="Derived series">derived series</a> and <a href="Lower_central_series" class="mw-redirect" title="Lower central series">lower central series</a> are descending series, while the <a href="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">Noetherian groups, Artinian groups</h3></div>

<p>A group that satisfies the <a href="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="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="Artin_group" class="mw-redirect" title="Artin group">Artin groups</a>), by analogy with <a href="Noetherian_ring" title="Noetherian ring">Noetherian rings</a> and <a href="Artinian_ring" title="Artinian ring">Artinian rings</a>. The ACC is equivalent to the <b>maximal condition</b>: every <a href="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="Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a>, and unlike for <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>, a group can be Artinian but not Noetherian, such as the <a href="Pr%C3%BCfer_group" title="Prüfer group">Prüfer group</a>. Every finite group is clearly Noetherian and Artinian.

</p><p><a href="Group_homomorphism" title="Group homomorphism">Homomorphic</a> <a href="Image_(mathematics)" title="Image (mathematics)">images</a> and subgroups of Noetherian groups are Noetherian, and an <a href="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="Finitely_generated_group" title="Finitely generated group">finitely generated</a>, which is stronger than the group itself being finitely generated: the <a href="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="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></div>

<p>Infinite subgroup series can also be defined and arise naturally, in which case the specific (<a href="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 mwe-math-element-inline"><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}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq G}</annotation>

</semantics>

</math></span><img src="./d999d6a344dba5bc6354fc07f3b3b8e6f8403168.svg" 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}" loading="lazy"></span> where the <span class="mwe-math-element mwe-math-element-inline"><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}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation>

</semantics>

</math></span><img src="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" 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}}" loading="lazy"></span> are indexed by the <a href="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="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>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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="./ce96dc347c5f4f09d07deeb93a36b510d959958c.svg" 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}" loading="lazy"></span></dd></dl>

<p>Given a recursive formula for producing a series, one can define a transfinite series by <a href="Transfinite_recursion" class="mw-redirect" title="Transfinite recursion">transfinite recursion</a> by defining the series at <a href="Limit_ordinal" title="Limit ordinal">limit ordinals</a> by

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{\lambda }:=\bigcup _{\alpha <\lambda }A_{\alpha }}</annotation>

</semantics>

</math></span><img src="./6690f82cb6f7ceccc75a34caec4048f931d775cb.svg" 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 }}" loading="lazy"></span> (for ascending series) or <span class="mwe-math-element mwe-math-element-inline"><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 }}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{\lambda }:=\bigcap _{\alpha <\lambda }A_{\alpha }}</annotation>

</semantics>

</math></span><img src="./92fd19c72cc852b56fbfbad152675911a27bc159.svg" 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 }}" loading="lazy"></span> (for descending series). Fundamental examples of this construction are the transfinite <a href="Lower_central_series" class="mw-redirect" title="Lower central series">lower central series</a> and <a href="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. For instance, one can define but rarely sees naturally occurring bi-infinite subgroup series (series indexed by the <a href="Integer" title="Integer">integers</a>):

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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="./07bc7fbaeb4429c086276f732511a558888a53e2.svg" 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}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Comparison_of_series">Comparison of series</h2></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="Bijection" title="Bijection">bijection</a> between the sets of their factor groups such that the corresponding factor groups are <a href="Group_isomorphism" title="Group isomorphism">isomorphic</a>. Refinement gives a <a href="Partial_order" class="mw-redirect" title="Partial order">partial order</a> on series, up to equivalence, and they form a <a href="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="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></div>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style>

<div class="mw-heading mw-heading3"><h3 id="Maximal_series">Maximal series</h3></div>

<ul><li>A <b><a href="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="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="Simple_group" title="Simple group">simple</a>.</dd></dl>

<ul><li>A <b><a href="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></div>

<ul><li>A <b><a href="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="Abelian_group" title="Abelian group">abelian</a>.</li>

<li>A <b><a href="Nilpotent_series" class="mw-redirect" title="Nilpotent series">nilpotent series</a></b> is a subnormal series such that successive quotients are <a href="Nilpotent_group" title="Nilpotent group">nilpotent</a>.</li></ul>

<dl><dd>A nilpotent series exists if and only if the group is <a href="Solvable_group" title="Solvable group">solvable</a>.</dd></dl>

<ul><li>A <b><a href="Central_series" title="Central series">central series</a></b> is a subnormal series such that successive quotients are <a href="Center_(group)" class="mw-redirect" title="Center (group)">central</a>, i.e. given the above series, <span class="mwe-math-element mwe-math-element-inline"><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})}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i+1}/A_{i}\subseteq Z(G/A_{i})}</annotation>

</semantics>

</math></span><img src="./0961a78d8d1e68d66a60394260b9287bae6a45ac.svg" 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})}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=0,1,\ldots ,n-2}</annotation>

</semantics>

</math></span><img src="./4120cb6112ac26d9c6d59f5965636abc4faf235c.svg" 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}" loading="lazy"></span>.</li></ul>

<dl><dd>A central series exists if and only if the group is <a href="Nilpotent_group" title="Nilpotent group">nilpotent</a>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Functional_series">Functional series</h3></div>

<p>Some subgroup series are defined functionally, in terms of subgroups such as the center and operations such as the commutator. These include:

</p>

<ul><li><a href="Lower_central_series" class="mw-redirect" title="Lower central series">Lower central series</a></li>

<li><a href="Upper_central_series" class="mw-redirect" title="Upper central series">Upper central series</a></li>

<li><a href="Derived_series" class="mw-redirect" title="Derived series">Derived series</a></li>

<li><a href="Lower_Fitting_series" class="mw-redirect" title="Lower Fitting series">Lower Fitting series</a></li>

<li><a href="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></div>

<p>There are series coming from subgroups of prime power order or prime power index, related to ideas such as <a href="Sylow_subgroup" class="mw-redirect" title="Sylow subgroup">Sylow subgroups</a>.

</p>

<ul><li><a href="Lower_p-series" class="mw-redirect" title="Lower p-series">Lower <i>p</i>-series</a></li>

<li><a href="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="Subgroup_method">Subgroup method</h2></div>

<p>The <b>subgroup method</b> is an algorithm used in the <a href="Mathematical" class="mw-redirect" title="Mathematical">mathematical</a> field of <a href="Group_theory" title="Group theory">group theory</a>. It is used to find the <a href="Word_(group_theory)" title="Word (group theory)">word</a> of an element. It doesn't always return the minimal word, but it can return optimal words based on the series of subgroups that is used. The code looks like this:

</p>

<pre><b>function</b> operate(element, generator)

<b>function</b> subgroup(g)

sequence := (set of subgroups that will be used, depending on the method.)

word := []

<b>for</b> subgroup in sequence

coset_representatives := []

<b>for</b> operation in coset_representatives

<b>if</b> <b>operate</b>(g, operation) is in the next subgroup <b>then</b>

append operation onto word

g = <b>operate</b>(g, operation)

<b>break</b>

<b>return</b> word

</pre>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

/* end https://en.wikipedia.org/ */

</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>: <span class="nowrap">343–</span>346.</cite> (English translation of <i>Dokl. Akad. Nauk SSSR</i>, <b>245</b>, 785–787)</span>

</li>

<cite id="CITEREFSharipov2009" class="citation arxiv cs1">Sharipov, R.A. (2009). "Transfinite normal and composition series of groups". <a href="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>

</li>

</ol></div></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-03" href="https://en.wikipedia.org/wiki/?title=Subgroup_series&oldid=1293723222">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>