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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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">For general description of the topic, see <a href="/wiki/Group_theory" title="Group theory">group theory</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">list of group theory topics</a></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Appendix:Glossary_of_group_theory" class="extiw" title="wiktionary:Appendix:Glossary of group theory">Appendix:Glossary of group theory</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output 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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 class="mw-selflink selflink">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 mw-collapsed"><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 href="/wiki/Free_group" title="Free group">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> <p>A <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> is a set together with an <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a> operation that admits an <a href="/wiki/Identity_element" title="Identity element">identity element</a> and such that there exists an <a href="/wiki/Inverse_element" title="Inverse element">inverse</a> for every element. </p><p>Throughout this glossary, we use <span class="texhtml"><i>e</i></span> to denote the identity element of a group. </p> <div class="noprint"><div role="navigation" id="toc" class="toc plainlinks" aria-label="Contents" style="text-align:left;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><div class="hlist"> <div style="margin:auto;"> <ul><li><a href="#A">A</a></li> <li><a href="#C">C</a></li> <li><a href="#D">D</a></li> <li><a href="#F">F</a></li> <li><a href="#G">G</a></li> <li><a href="#H">H</a></li> <li><a href="#I">I</a></li> <li><a href="#L">L</a></li> <li><a href="#N">N</a></li> <li><a href="#O">O</a></li> <li><a href="#P">P</a></li> <li><a href="#Q">Q</a></li> <li><a href="#R">R</a></li> <li><a href="#S">S</a></li> <li><a href="#T">T</a> </li></ul> <p class="mw-empty-elt"> </p> <ul><li><a href="#See_also">See also</a></li></ul> </div></div></div></div> <div class="mw-heading mw-heading2"><h2 id="A">A</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=1" title="Edit section: A"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1228772891">.mw-parser-output .glossary dt{margin-top:0.4em}.mw-parser-output .glossary dt+dt{margin-top:-0.2em}.mw-parser-output .glossary .templatequote{margin-top:0;margin-bottom:-0.5em}</style> <dl class="glossary"> <dt id="abelian_group"><dfn>abelian group</dfn></dt> <dd>A group <span class="texhtml">(<i>G</i>, •)</span> is <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> if <span class="texhtml">•</span> is commutative, i.e. <span class="texhtml"><i>g</i> • <i>h</i> = <i>h</i> • <i>g</i></span> for all <span class="texhtml"><i>g</i>, <i>h</i> ∈ <i>G</i></span>. Likewise, a group is <i>nonabelian</i> if this relation fails to hold for any pair <span class="texhtml"><i>g</i>, <i>h</i> ∈ <i>G</i></span>.</dd> <dt id="ascendant_subgroup"><dfn>ascendant subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> <span class="texhtml"><i>H</i></span> of a group <span class="texhtml"><i>G</i></span> is <a href="/wiki/Ascendant_subgroup" title="Ascendant subgroup">ascendant</a> if there is an ascending <a href="#subgroup_series"><span title="See entry on this page at § subgroup series" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup series</span></a> starting from <span class="texhtml"><i>H</i></span> and ending at <span class="texhtml"><i>G</i></span>, such that every term in the series is a <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> of its successor. The series may be infinite. If the series is finite, then the subgroup is <a href="#subnormal_subgroup"><span title="See entry on this page at § subnormal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subnormal</span></a>.</dd> <dt id="automorphism"><dfn>automorphism</dfn></dt> <dd>An <a href="/wiki/Group_automorphism" class="mw-redirect" title="Group automorphism">automorphism</a> of a group is an <a href="#isomorphism"><span title="See entry on this page at § isomorphism" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">isomorphism</span></a> of the group to itself.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="C">C</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=2" title="Edit section: C"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="center_of_a_group"><dfn>center of a group</dfn></dt> <dd>The <a href="/wiki/Center_of_a_group" class="mw-redirect" title="Center of a group">center of a group</a> <span class="texhtml"><i>G</i></span>, denoted <span class="texhtml">Z(<i>G</i>)</span>, is the set of those group elements that commute with all elements of <span class="texhtml"><i>G</i></span>, that is, the set of all <span class="texhtml"><i>h</i> ∈ <i>G</i></span> such that <span class="texhtml"><i>hg</i> = <i>gh</i></span> for all <span class="texhtml"><i>g</i> ∈ <i>G</i></span>. <span class="texhtml">Z(<i>G</i>)</span> is always a <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> of <span class="texhtml"><i>G</i></span>. A group <span class="texhtml"><i>G</i></span> is <a href="#abelian_group"><span title="See entry on this page at § Abelian group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">abelian</span></a> if and only if <span class="texhtml">Z(<i>G</i>) = <i>G</i></span>.</dd> <dt id="centerless_group"><dfn>centerless group</dfn></dt> <dd>A group <span class="texhtml"><i>G</i></span> is centerless if its <a href="#center_of_a_group"><span title="See entry on this page at § center of a group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">center</span></a> <span class="texhtml">Z(<i>G</i>)</span> is <a href="#trivial_group"><span title="See entry on this page at § trivial group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">trivial</span></a>.</dd> <dt id="central_subgroup"><dfn>central subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> of a group is a <a href="/wiki/Central_subgroup" title="Central subgroup">central subgroup</a> of that group if it lies inside the <a href="#center_of_a_group"><span title="See entry on this page at § center of a group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">center of the group</span></a>.</dd> <dt id="centralizer"><dfn>centralizer</dfn></dt> <dd>For a subset <span class="texhtml"><i>S</i></span> of a group <span class="texhtml"><i>G</i></span>, the <a href="/wiki/Centralizer_and_normalizer" title="Centralizer and normalizer">centralizer</a> of <span class="texhtml"><i>S</i></span> in <span class="texhtml"><i>G</i></span>, denoted <span class="texhtml">C<sub><i>G</i></sub>(<i>S</i>)</span>, is the subgroup of <span class="texhtml"><i>G</i></span> defined by <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 \mathrm {C} _{G}(S)=\{g\in G\mid gs=sg{\text{ for all }}s\in S\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo>∣</mo> <mi>g</mi> <mi>s</mi> <mo>=</mo> <mi>s</mi> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {C} _{G}(S)=\{g\in G\mid gs=sg{\text{ for all }}s\in S\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a88e32865958f7ce0dabdc8c528fde9f6029b2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.227ex; height:2.843ex;" alt="{\displaystyle \mathrm {C} _{G}(S)=\{g\in G\mid gs=sg{\text{ for all }}s\in S\}.}"></span></dd></dl></dd></dl> <dt id="characteristic_subgroup"><dfn>characteristic subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> of a group is a <a href="/wiki/Characteristic_subgroup" title="Characteristic subgroup">characteristic subgroup</a> of that group if it is mapped to itself by every <a href="#automorphism"><span title="See entry on this page at § automorphism" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">automorphism</span></a> of the parent group.</dd> <dt id="characteristically_simple_group"><dfn>characteristically simple group</dfn></dt> <dd>A group is said to be <a href="/wiki/Characteristically_simple_group" title="Characteristically simple group">characteristically simple</a> if it has no proper nontrivial <a href="#characteristic_subgroup"><span title="See entry on this page at § characteristic subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">characteristic subgroups</span></a>.</dd> <dt id="class_function"><dfn>class function</dfn></dt> <dd>A <a href="/wiki/Class_function" title="Class function">class function</a> on a group <span class="texhtml"><i>G</i></span> is a function that it is constant on the <a href="#conjugacy_class"><span title="See entry on this page at § conjugacy class" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">conjugacy classes</span></a> of <span class="texhtml"><i>G</i></span>.</dd> <dt id="class_number"><dfn>class number</dfn></dt> <dd>The <a href="/wiki/Class_number_(group_theory)" class="mw-redirect" title="Class number (group theory)">class number</a> of a group is the number of its <a href="#conjugacy_class"><span title="See entry on this page at § conjugacy class" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">conjugacy classes</span></a>.</dd> <dt id="commutator"><dfn>commutator</dfn></dt> <dd>The <a href="/wiki/Commutator_(group_theory)" class="mw-redirect" title="Commutator (group theory)">commutator</a> of two elements <span class="texhtml"><i>g</i></span> and <span class="texhtml"><i>h</i></span> of a group <span class="texhtml"><i>G</i></span> is the element <span class="texhtml">[<i>g</i>, <i>h</i>] = <i>g</i><sup>−1</sup><i>h</i><sup>−1</sup><i>gh</i></span>. Some authors define the commutator as <span class="texhtml">[<i>g</i>, <i>h</i>] = <i>ghg</i><sup>−1</sup><i>h</i><sup>−1</sup></span> instead. The commutator of two elements <span class="texhtml"><i>g</i></span> and <span class="texhtml"><i>h</i></span> is equal to the group's identity if and only if <span class="texhtml"><i>g</i></span> and <span class="texhtml"><i>h</i></span> commutate, that is, if and only if <span class="texhtml"><i>gh</i> = <i>hg</i></span>.</dd> <dt id="commutator_subgroup"><dfn>commutator subgroup</dfn></dt> <dd>The <a href="/wiki/Commutator_subgroup" title="Commutator subgroup">commutator subgroup</a> or derived subgroup of a group is the subgroup <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by all the <a href="#commutator"><span title="See entry on this page at § commutator" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">commutators</span></a> of the group.</dd> <dt id="composition_series"><dfn>composition series</dfn></dt> <dd>A <a href="/wiki/Composition_series_(group_theory)" class="mw-redirect" title="Composition series (group theory)">composition series</a> of a group <span class="texhtml"><i>G</i></span> is a <a href="/wiki/Subnormal_series" class="mw-redirect" title="Subnormal series">subnormal series</a> of finite length <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=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>◃</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>◃</mo> <mo>⋯</mo> <mo>◃</mo> <msub> <mi>H</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=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed70241b071404c74a310da0e1f00816ad6b26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.261ex; height:2.509ex;" alt="{\displaystyle 1=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G,}"></span></dd></dl> with strict inclusions, such that each <span class="texhtml"><i>H</i><sub><i>i</i></sub></span> is a maximal strict <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> of <span class="texhtml"><i>H</i><sub><i>i</i>+1</sub></span>. Equivalently, a composition series is a subnormal series such that each <a href="#factor_group"><span title="See entry on this page at § factor group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">factor group</span></a> <span class="texhtml"><i>H</i><sub><i>i</i>+1</sub> / <i>H</i><sub><i>i</i></sub></span> is <a href="#simple_group"><span title="See entry on this page at § simple group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">simple</span></a>. The factor groups are called composition factors.</dd> <dt id="conjugacy-closed_subgroup"><dfn>conjugacy-closed subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> of a group is said to be <a href="/wiki/Conjugacy-closed_subgroup" title="Conjugacy-closed subgroup">conjugacy-closed</a> if any two elements of the subgroup that are <a href="#conjugate_elements"><span title="See entry on this page at § conjugate elements" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">conjugate</span></a> in the group are also conjugate in the subgroup.</dd> <dt id="conjugacy_class"><dfn>conjugacy class</dfn></dt> <dd>The <a href="/wiki/Conjugacy_classes" class="mw-redirect" title="Conjugacy classes">conjugacy classes</a> of a group <span class="texhtml"><i>G</i></span> are those subsets of <span class="texhtml"><i>G</i></span> containing group elements that are <a href="#conjugate_elements"><span title="See entry on this page at § conjugate elements" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">conjugate</span></a> with each other.</dd> <dt id="conjugate_elements"><dfn>conjugate elements</dfn></dt> <dd>Two elements <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> of a group <span class="texhtml"><i>G</i></span> are <a href="/wiki/Conjugate_(group_theory)" class="mw-redirect" title="Conjugate (group theory)">conjugate</a> if there exists an element <span class="texhtml"><i>g</i> ∈ <i>G</i></span> such that <span class="texhtml"><i>g</i><sup>−1</sup><i>xg</i> = <i>y</i></span>. The element <span class="texhtml"><i>g</i><sup>−1</sup><i>xg</i></span>, denoted <span class="texhtml"><i>x</i><sup><i>g</i></sup></span>, is called the conjugate of <span class="texhtml"><i>x</i></span> by <span class="texhtml"><i>g</i></span>. Some authors define the conjugate of <span class="texhtml"><i>x</i></span> by <span class="texhtml"><i>g</i></span> as <span class="texhtml"><i>gxg</i><sup>−1</sup></span>. This is often denoted <span class="texhtml"><sup><i>g</i></sup><i>x</i></span>. Conjugacy is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>. Its <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> are called <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy classes</a>.</dd> <dt id="conjugate_subgroups"><dfn>conjugate subgroups</dfn></dt> <dd>Two subgroups <span class="texhtml"><i>H</i><sub>1</sub></span> and <span class="texhtml"><i>H</i><sub>2</sub></span> of a group <span class="texhtml"><i>G</i></span> are <a href="/wiki/Conjugate_subgroups" class="mw-redirect" title="Conjugate subgroups">conjugate subgroups</a> if there is a <span class="texhtml"><i>g</i> ∈ <i>G</i></span> such that <span class="texhtml"><i>gH</i><sub>1</sub><i>g</i><sup>−1</sup> = <i>H</i><sub>2</sub></span>.</dd> <dt id="contranormal_subgroup"><dfn>contranormal subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> of a group <span class="texhtml"><i>G</i></span> is a <a href="/wiki/Contranormal_subgroup" title="Contranormal subgroup">contranormal subgroup</a> of <span class="texhtml"><i>G</i></span> if its <a href="#normal_closure"><span title="See entry on this page at § normal closure" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal closure</span></a> is <span class="texhtml"><i>G</i></span> itself.</dd> <dt id="cyclic_group"><dfn>cyclic group</dfn></dt> <dd>A <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> is a group that is <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by a single element, that is, a group such that there is an element <span class="texhtml"><i>g</i></span> in the group such that every other element of the group may be obtained by repeatedly applying the group operation to <span class="texhtml"><i>g</i></span> or its inverse.</dd> <div class="mw-heading mw-heading2"><h2 id="D">D</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=3" title="Edit section: D"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="derived_subgroup"><dfn>derived subgroup</dfn></dt> <dd>Synonym for <a href="#commutator_subgroup"><span title="See entry on this page at § commutator subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">commutator subgroup</span></a>.</dd> <dt id="direct_product"><dfn>direct product</dfn></dt> <dd>The <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a> of two groups <span class="texhtml"><i>G</i></span> and <span class="texhtml"><i>H</i></span>, denoted <span class="texhtml"><i>G</i> × <i>H</i></span>, is the <a href="/wiki/Cartesian_product" title="Cartesian product">cartesian product</a> of the underlying sets of <span class="texhtml"><i>G</i></span> and <span class="texhtml"><i>H</i></span>, equipped with a component-wise defined binary operation <span class="texhtml">(<i>g</i><sub>1</sub>, <i>h</i><sub>1</sub>) · (<i>g</i><sub>2</sub>, <i>h</i><sub>2</sub>) = (<i>g</i><sub>1</sub> ⋅ <i>g</i><sub>2</sub>, <i>h</i><sub>1</sub> ⋅ <i>h</i><sub>2</sub>)</span>. With this operation, <span class="texhtml"><i>G</i> × <i>H</i></span> itself forms a group.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="E">E</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=4" title="Edit section: E"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="exponent_of_a_group"><dfn>exponent of a group</dfn></dt> <dd>The exponent of a group <span class="texhtml"><i>G</i></span> is the smallest positive integer <span class="texhtml"><i>n</i></span> such that <span class="texhtml"><i>g</i><sup><i>n</i></sup> = <i>e</i></span> for all <span class="texhtml"><i>g</i> ∈ <i>G</i></span>. It is the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> of the <a href="#order_of_a_group"><span title="See entry on this page at § order of a group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">orders</span></a> of all elements in the group. If no such positive integer exists, the exponent of the group is said to be infinite.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="F">F</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=5" title="Edit section: F"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="factor_group"><dfn>factor group</dfn></dt> <dd>Synonym for <a href="#quotient_group"><span title="See entry on this page at § quotient group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">quotient group</span></a>.</dd> <dt id="fc-group"><dfn>FC-group</dfn></dt> <dd>A group is an <a href="/wiki/FC-group" title="FC-group">FC-group</a> if every <a href="#conjugacy_class"><span title="See entry on this page at § conjugacy class" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">conjugacy class</span></a> of its elements has finite cardinality.</dd> <dt id="finite_group"><dfn>finite group</dfn></dt> <dd>A <a href="/wiki/Finite_group" title="Finite group">finite group</a> is a group of finite <a href="#order_of_a_group"><span title="See entry on this page at § order of a group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">order</span></a>, that is, a group with a finite number of elements.</dd> <dt id="finitely_generated_group"><dfn>finitely generated group</dfn></dt> <dd>A group <span class="texhtml"><i>G</i></span> is <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated</a> if there is a finite <a href="#generating_set"><span title="See entry on this page at § generating set" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">generating set</span></a>, that is, if there is a finite set <span class="texhtml"><i>S</i></span> of elements of <span class="texhtml mvar" style="font-style:italic;">G</span> such that every element of <span class="texhtml"><i>G</i></span> can be written as the combination of finitely many elements of <span class="texhtml"><i>S</i></span> and of inverses of elements of <span class="texhtml"><i>S</i></span>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="G">G</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=6" title="Edit section: G"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="generating_set"><dfn>generating set</dfn></dt> <dd>A <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generating set</a> of a group <span class="texhtml"><i>G</i></span> is a subset <span class="texhtml"><i>S</i></span> of <span class="texhtml"><i>G</i></span> such that every element of <span class="texhtml"><i>G</i></span> can be expressed as a combination (under the group operation) of finitely many elements of <span class="texhtml"><i>S</i></span> and inverses of elements of <span class="texhtml">S</span>. Given a subset <span class="texhtml"><i>S</i></span> of <span class="texhtml"><i>G</i></span>. We denote by <span class="texhtml"><span class="nowrap">⟨<i>S</i>⟩</span></span> the smallest subgroup of <span class="texhtml"><i>G</i></span> containing <span class="texhtml"><i>S</i></span>. <span class="texhtml"><span class="nowrap">⟨<i>S</i>⟩</span></span> is called the subgroup of <span class="texhtml"><i>G</i></span> generated by <span class="texhtml"><i>S</i></span>.</dd> <dt id="group_automorphism"><dfn>group automorphism</dfn></dt> <dd>See <a href="#automorphism"><span title="See entry on this page at § automorphism" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">automorphism</span></a>.</dd> <dt id="group_homomorphism"><dfn>group homomorphism</dfn></dt> <dd>See <a href="#homomorphism"><span title="See entry on this page at § homomorphism" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">homomorphism</span></a>.</dd> <dt id="group_isomorphism"><dfn>group isomorphism</dfn></dt> <dd>See <a href="#isomorphism"><span title="See entry on this page at § isomorphism" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">isomorphism</span></a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="H">H</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=7" title="Edit section: H"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="homomorphism"><dfn>homomorphism</dfn></dt> <dd>Given two groups <span class="texhtml">(<i>G</i>, •)</span> and <span class="texhtml">(<i>H</i>, ·)</span>, a <a href="/wiki/Group_homomorphism" title="Group homomorphism">homomorphism</a> from <span class="texhtml"><i>G</i></span> to <span class="texhtml"><i>H</i></span> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml"><i>h</i> : <i>G</i> → <i>H</i></span> such that for all <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> in <span class="texhtml"><i>G</i></span>, <span class="texhtml"><i>h</i>(<i>a</i> • <i>b</i>) = <i>h</i>(<i>a</i>) · <i>h</i>(<i>b</i>)</span>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="I">I</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=8" title="Edit section: I"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="index_of_a_subgroup"><dfn>index of a subgroup</dfn></dt> <dd>The <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> of a <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> <span class="texhtml"><i>H</i></span> of a group <span class="texhtml"><i>G</i></span>, denoted <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>G</i> : <i>H</i></span>|</span> or <span class="texhtml">[<i>G</i> : <i>H</i>]</span> or <span class="texhtml">(<i>G</i> : <i>H</i>)</span>, is the number of <a href="/wiki/Coset" title="Coset">cosets</a> of <span class="texhtml"><i>H</i></span> in <span class="texhtml"><i>G</i></span>. For a <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> <span class="texhtml"><i>N</i></span> of a group <span class="texhtml"><i>G</i></span>, the index of <span class="texhtml"><i>N</i></span> in <span class="texhtml"><i>G</i></span> is equal to the <a href="#order_of_a_group"><span title="See entry on this page at § order of a group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">order</span></a> of the <a href="#quotient_group"><span title="See entry on this page at § quotient group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">quotient group</span></a> <span class="texhtml"><i>G</i> / <i>N</i></span>. For a <a href="#finite_group"><span title="See entry on this page at § finite group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">finite</span></a> subgroup <span class="texhtml"><i>H</i></span> of a finite group <span class="texhtml"><i>G</i></span>, the index of <span class="texhtml"><i>H</i></span> in <span class="texhtml"><i>G</i></span> is equal to the quotient of the orders of <span class="texhtml"><i>G</i></span> and <span class="texhtml"><i>H</i></span>.</dd> <dt id="isomorphism"><dfn>isomorphism</dfn></dt> <dd>Given two groups <span class="texhtml">(<i>G</i>, •)</span> and <span class="texhtml">(<i>H</i>, ·)</span>, an <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphism</a> between <span class="texhtml"><i>G</i></span> and <span class="texhtml"><i>H</i></span> is a <a href="/wiki/Bijection" title="Bijection">bijective</a> <a href="#homomorphism"><span title="See entry on this page at § homomorphism" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">homomorphism</span></a> from <span class="texhtml"><i>G</i></span> to <span class="texhtml"><i>H</i></span>, that is, a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Two groups are <i>isomorphic</i> if there exists a group isomorphism mapping from one to the other. Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="L">L</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=9" title="Edit section: L"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="lattice_of_subgroups"><dfn>lattice of subgroups</dfn></dt> <dd>The <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">lattice of subgroups</a> of a group is the <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> defined by its <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroups</span></a>, <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered</a> by <a href="/wiki/Set_inclusion" class="mw-redirect" title="Set inclusion">set inclusion</a>.</dd> <dt id="locally_cyclic_group"><dfn>locally cyclic group</dfn></dt> <dd>A group is <a href="/wiki/Locally_cyclic_group" title="Locally cyclic group">locally cyclic</a> if every <a href="#finitely_generated_group"><span title="See entry on this page at § finitely generated group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">finitely generated</span></a> subgroup is <a href="#cyclic_group"><span title="See entry on this page at § cyclic group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">cyclic</span></a>. Every cyclic group is locally cyclic, and every <a href="#finitely_generated_group"><span title="See entry on this page at § finitely generated group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">finitely-generated</span></a> locally cyclic group is cyclic. Every locally cyclic group is <a href="#abelian_group"><span title="See entry on this page at § abelian group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">abelian</span></a>. Every <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a>, every <a href="#quotient_group"><span title="See entry on this page at § quotient group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">quotient group</span></a> and every <a href="#group_homomorphism"><span title="See entry on this page at § group homomorphism" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">homomorphic</span></a> image of a locally cyclic group is locally cyclic.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="N">N</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=10" title="Edit section: N"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="no_small_subgroup"><dfn>no small subgroup</dfn></dt> <dd>A <a href="/wiki/Topological_group" title="Topological group">topological group</a> has <a href="/wiki/No_small_subgroup" title="No small subgroup">no small subgroup</a> if there exists a neighborhood of the identity element that does not contain any nontrivial subgroup.</dd> <dt id="normal_closure"><dfn>normal closure</dfn></dt> <dd>The <a href="/wiki/Normal_closure_(group_theory)" title="Normal closure (group theory)">normal closure</a> of a subset <span class="texhtml"><i>S</i></span> of a group <span class="texhtml"><i>G</i></span> is the intersection of all <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroups</span></a> of <span class="texhtml"><i>G</i></span> that contain <span class="texhtml"><i>S</i></span>.</dd> <dt id="normal_core"><dfn>normal core</dfn></dt> <dd>The <a href="/wiki/Normal_core" class="mw-redirect" title="Normal core">normal core</a> of a <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> <span class="texhtml"><i>H</i></span> of a group <span class="texhtml"><i>G</i></span> is the largest <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> of <span class="texhtml"><i>G</i></span> that is contained in <span class="texhtml"><i>H</i></span>.</dd> <dt id="normal_series"><dfn>normal series</dfn></dt> <dd>A <a href="/wiki/Normal_series" class="mw-redirect" title="Normal series">normal series</a> of a group <span class="texhtml"><i>G</i></span> is a sequence of <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroups</span></a> of <span class="texhtml"><i>G</i></span> such that each element of the sequence is a normal subgroup of the next element: <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> </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/6e9db9ae4c211193cfb9db08a0f9bb4175a96fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.05ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\triangleleft A_{1}\triangleleft \cdots \triangleleft A_{n}=G}"></span></dd></dl> with <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 A_{i}\triangleleft G}"> <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> <mo>◃</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}\triangleleft G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef32fe22faef5e4be2c5f65a930f0a61ebee7de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.564ex; height:2.509ex;" alt="{\displaystyle A_{i}\triangleleft G}"></span>.</dd></dl></dd></dl> <dt id="normal_subgroup"><dfn>normal subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> <span class="texhtml"><i>N</i></span> of a group <span class="texhtml"><i>G</i></span> is <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal</a> in <span class="texhtml"><i>G</i></span> (denoted <span class="texhtml"><i>N</i> ◅ <i>G</i></span>) if the <a href="#conjugate_elements"><span title="See entry on this page at § conjugate elements" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">conjugation</span></a> of an element <span class="texhtml"><i>n</i></span> of <span class="texhtml"><i>N</i></span> by an element <span class="texhtml"><i>g</i></span> of <span class="texhtml"><i>G</i></span> is always in <span class="texhtml"><i>N</i></span>, that is, if for all <span class="texhtml"><i>g</i> ∈ <i>G</i></span> and <span class="texhtml"><i>n</i> ∈ <i>N</i></span>, <span class="texhtml"><i>gng</i><sup>−1</sup> ∈ <i>N</i></span>. A normal subgroup <span class="texhtml"><i>N</i></span> of a group <span class="texhtml"><i>G</i></span> can be used to construct the <a href="#quotient_group"><span title="See entry on this page at § quotient group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">quotient group</span></a> <span class="texhtml"><i>G</i> / <i>N</i></span>.</dd> <dt id="normalizer"><dfn>normalizer</dfn></dt> <dd>For a subset <span class="texhtml"><i>S</i></span> of a group <span class="texhtml"><i>G</i></span>, the <a href="/wiki/Centralizer_and_normalizer" title="Centralizer and normalizer">normalizer</a> of <span class="texhtml"><i>S</i></span> in <span class="texhtml"><i>G</i></span>, denoted <span class="texhtml">N<sub><i>G</i></sub>(<i>S</i>)</span>, is the subgroup of <span class="texhtml"><i>G</i></span> defined by <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 \mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo>∣</mo> <mi>g</mi> <mi>S</mi> <mo>=</mo> <mi>S</mi> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038ee9ce297092d8ad882e97a4a2e3cdd2f4f3ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.695ex; height:2.843ex;" alt="{\displaystyle \mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}.}"></span></dd></dl></dd> <div class="mw-heading mw-heading2"><h2 id="O">O</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=11" title="Edit section: O"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="orbit"><dfn>orbit</dfn></dt> <dd>Consider a group <span class="texhtml"><i>G</i></span> acting on a set <span class="texhtml"><i>X</i></span>. The <a href="/wiki/Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">orbit</a> of an element <span class="texhtml"><i>x</i></span> in <span class="texhtml"><i>X</i></span> is the set of elements in <span class="texhtml"><i>X</i></span> to which <span class="texhtml"><i>x</i></span> can be moved by the elements of <span class="texhtml"><i>G</i></span>. The orbit of <span class="texhtml"><i>x</i></span> is denoted by <span class="texhtml"><i>G</i> ⋅ <i>x</i></span></dd> <dt id="order_of_a_group"><dfn>order of a group</dfn></dt> <dd>The <a href="/wiki/Order_of_a_group" class="mw-redirect" title="Order of a group">order of a group</a> <span class="texhtml">(<i>G</i>, •)</span> is the <a href="/wiki/Cardinal_number" title="Cardinal number">cardinality</a> (i.e. number of elements) of <span class="texhtml">G</span>. A group with finite order is called a <a href="/wiki/Finite_group" title="Finite group">finite group</a>.</dd> <dt id="order_of_a_group_element"><dfn>order of a group element</dfn></dt> <dd>The <a href="/wiki/Order_of_a_group_element" class="mw-redirect" title="Order of a group element">order of an element</a> <span class="texhtml"><i>g</i></span> of a group <span class="texhtml"><i>G</i></span> is the smallest <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive</a> <a href="/wiki/Integer" title="Integer">integer</a> <span class="texhtml"><i>n</i></span> such that <span class="texhtml"><i>g</i><sup><i>n</i></sup> = <i>e</i></span>. If no such integer exists, then the order of <span class="texhtml"><i>g</i></span> is said to be infinite. The order of a finite group is <a href="/wiki/Divisible" class="mw-redirect" title="Divisible">divisible</a> by the order of every element.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="P">P</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=12" title="Edit section: P"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="perfect_core"><dfn>perfect core</dfn></dt> <dd>The <a href="/wiki/Perfect_core" title="Perfect core">perfect core</a> of a group is its largest <a href="#perfect_group"><span title="See entry on this page at § perfect group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">perfect</span></a> subgroup.</dd> <dt id="perfect_group"><dfn>perfect group</dfn></dt> <dd>A <a href="/wiki/Perfect_group" title="Perfect group">perfect group</a> is a group that is equal to its own <a href="#commutator_subgroup"><span title="See entry on this page at § commutator subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">commutator subgroup</span></a>.</dd> <dt id="periodic_group"><dfn>periodic group</dfn></dt> <dd>A group is <a href="/wiki/Periodic_group" class="mw-redirect" title="Periodic group">periodic</a> if every group element has finite <a href="#order_of_a_group_element"><span title="See entry on this page at § order of a group element" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">order</span></a>. Every <a href="/wiki/Finite_group" title="Finite group">finite group</a> is periodic.</dd> <dt id="permutation_group"><dfn>permutation group</dfn></dt> <dd>A <a href="/wiki/Permutation_group" title="Permutation group">permutation group</a> is a group whose elements are <a href="/wiki/Permutation" title="Permutation">permutations</a> of a given <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml"><i>M</i></span> (the <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective functions</a> from set <span class="texhtml"><i>M</i></span> to itself) and whose <a href="/wiki/Group_operation" class="mw-redirect" title="Group operation">group operation</a> is the <a href="/wiki/Function_composition" title="Function composition">composition</a> of those permutations. The group consisting of all permutations of a set <span class="texhtml"><i>M</i></span> is the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> of <span class="texhtml"><i>M</i></span>.</dd> <dt id="''p''-group"><dfn><i>p</i>-group</dfn></dt> <dd>If <span class="texhtml"><i>p</i></span> is a <a href="/wiki/Prime_number" title="Prime number">prime number</a>, then a <a href="/wiki/P-group" title="P-group"><span class="texhtml"><i>p</i></span>-group</a> is one in which the order of every element is a power of <span class="texhtml"><i>p</i></span>. A finite group is a <span class="texhtml"><i>p</i></span>-group if and only if the <a href="#order_of_a_group"><span title="See entry on this page at § order of a group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">order</span></a> of the group is a power of <span class="texhtml"><i>p</i></span>.</dd> <dt id="''p''-subgroup"><dfn><i>p</i>-subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> that is also a <a href="#p-group"><span title="See entry on this page at § ''p''-group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;"><span class="texhtml"><i>p</i></span>-group</span></a>. The study of <span class="texhtml"><i>p</i></span>-subgroups is the central object of the <a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Q">Q</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=13" title="Edit section: Q"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="quotient_group"><dfn>quotient group</dfn></dt> <dd>Given a group <span class="texhtml"><i>G</i></span> and a <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> <span class="texhtml"><i>N</i></span> of <span class="texhtml"><i>G</i></span>, the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> is the set <span class="texhtml"><i>G</i> / <i>N</i></span> of <a href="/wiki/Left_coset" class="mw-redirect" title="Left coset">left cosets</a> <span class="texhtml">{<i>aN</i> : <i>a</i> ∈ <i>G</i>}</span> together with the operation <span class="texhtml"><i>aN</i> • <i>bN</i> = <i>abN</i></span>. The relationship between normal subgroups, homomorphisms, and factor groups is summed up in the <a href="/wiki/Fundamental_theorem_on_homomorphisms" title="Fundamental theorem on homomorphisms">fundamental theorem on homomorphisms</a>.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="R">R</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=14" title="Edit section: R"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="real_element"><dfn>real element</dfn></dt> <dd>An element <span class="texhtml"><i>g</i></span> of a group <span class="texhtml"><i>G</i></span> is called a <a href="/wiki/Real_element" title="Real element">real element</a> of <span class="texhtml"><i>G</i></span> if it belongs to the same <a href="#conjugacy_class"><span title="See entry on this page at § conjugacy class" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">conjugacy class</span></a> as its inverse, that is, if there is a <span class="texhtml"><i>h</i></span> in <span class="texhtml"><i>G</i></span> with <span class="texhtml"><i>g</i><sup><i>h</i></sup> = <i>g</i><sup>−1</sup></span>, where <span class="texhtml"><i>g</i><sup><i>h</i></sup></span> is defined as <span class="texhtml"><i>h</i><sup>−1</sup><i>gh</i></span>. An element of a group <span class="texhtml"><i>G</i></span> is real if and only if for all <a href="/wiki/Group_representation" title="Group representation">representations</a> of <span class="texhtml"><i>G</i></span> the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of the corresponding matrix is a real number.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="S">S</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=15" title="Edit section: S"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="serial_subgroup"><dfn>serial subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> <span class="texhtml"><i>H</i></span> of a group <span class="texhtml"><i>G</i></span> is a <a href="/wiki/Serial_subgroup" title="Serial subgroup">serial subgroup</a> of <span class="texhtml"><i>G</i></span> if there is a chain <span class="texhtml"><i>C</i></span> of subgroups of <span class="texhtml"><i>G</i></span> from <span class="texhtml"><i>H</i></span> to <span class="texhtml"><i>G</i></span> such that for each pair of consecutive subgroups <span class="texhtml"><i>X</i></span> and <span class="texhtml"><i>Y</i></span> in <span class="texhtml"><i>C</i></span>, <span class="texhtml"><i>X</i></span> is a <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> of <span class="texhtml"><i>Y</i></span>. If the chain is finite, then <span class="texhtml"><i>H</i></span> is a <a href="#subnormal_subgroup"><span title="See entry on this page at § subnormal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subnormal subgroup</span></a> of <span class="texhtml"><i>G</i></span>.</dd> <dt id="simple_group"><dfn>simple group</dfn></dt> <dd>A <a href="/wiki/Simple_group" title="Simple group">simple group</a> is a <a href="#trivial_group"><span title="See entry on this page at § trivial group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">nontrivial group</span></a> whose only <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroups</span></a> are the trivial group and the group itself.</dd> <dt id="subgroup"><dfn>subgroup</dfn></dt> <dd>A <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of a group <span class="texhtml"><i>G</i></span> is a <a href="/wiki/Subset" title="Subset">subset</a> <span class="texhtml"><i>H</i></span> of the elements of <span class="texhtml"><i>G</i></span> that itself forms a group when equipped with the restriction of the <a href="/wiki/Group_operation" class="mw-redirect" title="Group operation">group operation</a> of <span class="texhtml"><i>G</i></span> to <span class="texhtml"><i>H</i> × <i>H</i></span>. A subset <span class="texhtml"><i>H</i></span> of a group <span class="texhtml"><i>G</i></span> is a subgroup of <span class="texhtml"><i>G</i></span> if and only if it is nonempty and <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under products and inverses, that is, if and only if for every <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> in <span class="texhtml"><i>H</i></span>, <span class="texhtml"><i>ab</i></span> and <span class="texhtml"><i>a</i><sup>−1</sup></span> are also in <span class="texhtml"><i>H</i></span>.</dd> <dt id="subgroup_series"><dfn>subgroup series</dfn></dt> <dd>A <a href="/wiki/Subgroup_series" title="Subgroup series">subgroup series</a> of a group <span class="texhtml"><i>G</i></span> is a sequence of <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroups</span></a> of <span class="texhtml"><i>G</i></span> such that each element in the series is a subgroup of the next element: <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> <mo>.</mo> </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/1d4ee117c021e508ea35f3cf7b1e65ea4adf7ff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.408ex; height:2.509ex;" alt="{\displaystyle 1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G.}"></span></dd></dl></dd></dl> <dt id="subnormal_subgroup"><dfn>subnormal subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> <span class="texhtml"><i>H</i></span> of a group <span class="texhtml"><i>G</i></span> is a <a href="/wiki/Subnormal_subgroup" title="Subnormal subgroup">subnormal subgroup</a> of <span class="texhtml"><i>G</i></span> if there is a finite chain of subgroups of the group, each one <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal</span></a> in the next, beginning at <span class="texhtml"><i>H</i></span> and ending at <span class="texhtml"><i>G</i></span>.</dd> <dt id="symmetric_group"><dfn>symmetric group</dfn></dt> <dd>Given a set <span class="texhtml"><i>M</i></span>, the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> of <span class="texhtml"><i>M</i></span> is the set of all <a href="/wiki/Permutation" title="Permutation">permutations</a> of <span class="texhtml"><i>M</i></span> (the set all <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective functions</a> from <span class="texhtml"><i>M</i></span> to <span class="texhtml"><i>M</i></span>) with the <a href="/wiki/Function_composition" title="Function composition">composition</a> of the permutations as group operation. The symmetric group of a <a href="/wiki/Finite_set" title="Finite set">finite set</a> of size <span class="texhtml"><i>n</i></span> is denoted <span class="texhtml">S<sub><i>n</i></sub></span>. (The symmetric groups of any two sets of the same size are <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a>.)</dd> <div class="mw-heading mw-heading2"><h2 id="T">T</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=16" title="Edit section: T"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1228772891"> <dl class="glossary"> <dt id="torsion_group"><dfn>torsion group</dfn></dt> <dd>Synonym for <a href="#periodic_group"><span title="See entry on this page at § periodic group" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">periodic group</span></a>.</dd> <dt id="transitively_normal_subgroup"><dfn>transitively normal subgroup</dfn></dt> <dd>A <a href="#subgroup"><span title="See entry on this page at § subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">subgroup</span></a> of a group is said to be <a href="/wiki/Transitively_normal_subgroup" title="Transitively normal subgroup">transitively normal</a> in the group if every <a href="#normal_subgroup"><span title="See entry on this page at § normal subgroup" class="glossary-link-internal" style="border-bottom:1px dashed #86a1ff;color:initial;">normal subgroup</span></a> of the subgroup is also normal in the whole group.</dd> <dt id="trivial_group"><dfn>trivial group</dfn></dt> <dd>A <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a> is a group consisting of a single element, namely the identity element of the group. All such groups are <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a>, and one often speaks of <i>the</i> trivial group.</dd> </dl> <div class="mw-heading mw-heading2"><h2 id="Basic_definitions">Basic definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=17" title="Edit section: Basic definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both subgroups and normal subgroups of a given group form a <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a> under inclusion of subsets; this property and some related results are described by the <a href="/wiki/Lattice_theorem" class="mw-redirect" title="Lattice theorem">lattice theorem</a>. </p><p><b><a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">Kernel</a> of a group homomorphism</b>. It is the <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimage</a> of the identity in the <a href="/wiki/Codomain" title="Codomain">codomain</a> of a group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa. </p><p><b><a href="/wiki/Direct_product" title="Direct product">Direct product</a></b>, <b><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">direct sum</a></b>, and <b><a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a></b> of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links for explanation. </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_groups">Types of groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Glossary_of_group_theory&action=edit&section=18" title="Edit section: Types of groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Finitely_generated_group" title="Finitely generated group">Finitely generated group</a></b>. If there exists a finite set <span class="texhtml"><i>S</i></span> such that <span class="texhtml"><span class="nowrap">⟨<i>S</i>⟩</span> = <i>G</i></span>, then <span class="texhtml"><i>G</i></span> is said to be <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">finitely generated</a>. If <span class="texhtml"><i>S</i></span> can be taken to have just one element, <span class="texhtml"><i>G</i></span> is a <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> of finite order, an <a href="/wiki/Infinite_cyclic_group" class="mw-redirect" title="Infinite cyclic group">infinite cyclic group</a>, or possibly a group <span class="texhtml">{<i>e</i>}</span> with just one element. </p><p><b><a href="/wiki/Simple_group" title="Simple group">Simple group</a></b>. Simple groups are those groups having only <span class="texhtml"><i>e</i></span> and themselves as <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroups</a>. The name is misleading because a simple group can in fact be very complex. An example is the <a href="/wiki/Monster_group" title="Monster group">monster group</a>, whose <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> is about 10<sup>54</sup>. Every finite group is built up from simple groups via <a href="/wiki/Extension_problem" class="mw-redirect" title="Extension problem">group extensions</a>, so the study of finite simple groups is central to the study of all finite groups. The finite simple groups are known and <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classified</a>. </p><p>The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a> p-groups. This can be extended to a complete classification of all <a href="/wiki/Finitely_generated_abelian_group" title="Finitely generated abelian group">finitely generated abelian groups</a>, that is all abelian groups that are <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by a finite set. </p><p>The situation is much more complicated for the non-abelian groups. </p><p><b><a href="/wiki/Free_group" title="Free group">Free group</a></b>. Given any set <span class="texhtml"><i>A</i></span>, one can define a group as the smallest group containing the <a href="/wiki/Free_semigroup" class="mw-redirect" title="Free semigroup">free semigroup</a> of <span class="texhtml"><i>A</i></span>. The group consists of the finite strings (words) that can be composed by elements from <span class="texhtml"><i>A</i></span>, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance <span class="texhtml">(<i>abb</i>) • (<i>bca</i>) = <i>abbbca</i></span>. </p><p>Every group <span class="texhtml">(<i>G</i>, •)</span> is basically a factor group of a free group generated by <span class="texhtml"><i>G</i></span>. Refer to <i><a href="/wiki/Presentation_of_a_group" title="Presentation of a group">Presentation of a group</a></i> for more explanation. One can then ask <a href="/wiki/Algorithm" title="Algorithm">algorithmic</a> questions about these presentations, such as: </p> <ul><li>Do these two presentations specify isomorphic groups?; or</li> <li>Does this presentation specify the trivial group?</li></ul> <p>The general case of this is the <a href="/wiki/Word_problem_for_groups" title="Word problem for groups">word problem</a>, and several of these questions are in fact unsolvable by any general algorithm. </p><p><b><a href="/wiki/General_linear_group" title="General linear group">General linear group</a></b>, denoted by <span class="texhtml">GL(<i>n</i>, <i>F</i>)</span>, is the group of <span class="texhtml"><i>n</i></span>-by-<span class="texhtml"><i>n</i></span> <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible matrices</a>, where the elements of the matrices are taken from a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml"><i>F</i></span> such as the real numbers or the complex numbers. </p><p><b><a href="/wiki/Group_representation" title="Group representation">Group representation</a></b> (not to be confused with the <i>presentation</i> of a group). A <i>group representation</i> is a homomorphism from a group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, which is much easier to study. </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=Glossary_of_group_theory&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Glossary_of_Lie_groups_and_Lie_algebras" title="Glossary of Lie groups and Lie algebras">Glossary of Lie groups and Lie algebras</a></li> <li><a href="/wiki/Glossary_of_ring_theory" title="Glossary of ring theory">Glossary of ring theory</a></li> <li><a href="/wiki/Composition_series" title="Composition series">Composition series</a></li> <li><a href="/wiki/Normal_series" class="mw-redirect" title="Normal series">Normal series</a></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=Glossary_of_group_theory&oldid=1237475250">https://en.wikipedia.org/w/index.php?title=Glossary_of_group_theory&oldid=1237475250</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:Group_theory" title="Category:Group theory">Group theory</a></li><li><a href="/wiki/Category:Glossaries_of_mathematics" title="Category:Glossaries of mathematics">Glossaries of mathematics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden category: <ul><li><a href="/wiki/Category:Wikipedia_glossaries_using_description_lists" title="Category:Wikipedia glossaries using description lists">Wikipedia glossaries using description lists</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 30 July 2024, at 00:05<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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