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<title>Simple group</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Simple group</span></span>

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</style><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="Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br><a href="Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"></span></span></td></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">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, #202122 ); 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="Subgroup" title="Subgroup">Subgroup</a></li>

<li><a href="Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li>

<li><a href="Group_action" title="Group action">Group action</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Quotient_group" title="Quotient group">Quotient group</a></li>

<li><a href="Semidirect_product" title="Semidirect product">(Semi-)</a><a href="Direct_product_of_groups" title="Direct product of groups">direct product</a></li>

<li><a href="Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li>

<li><a href="Free_product" title="Free product">Free product</a></li>

<li><a href="Wreath_product" title="Wreath product">Wreath product</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<i><a href="Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li>

<li><a href="Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul>

<li><a href="Finite_group" title="Finite group">finite</a></li>

<li><a href="Infinite_group" title="Infinite group">infinite</a></li>

<li><a href="Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li>

<li><a href="Multiplicative_group" title="Multiplicative group">multiplicative</a></li>

<li><a href="Additive_group" title="Additive group">additive</a></li>

<li><a href="Cyclic_group" title="Cyclic group">cyclic</a></li>

<li><a href="Abelian_group" title="Abelian group">abelian</a></li>

<li><a href="Dihedral_group" title="Dihedral group">dihedral</a></li>

<li><a href="Nilpotent_group" title="Nilpotent group">nilpotent</a></li>

<li><a href="Solvable_group" title="Solvable group">solvable</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li>

<li><a href="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="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, #202122 ); 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="Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li>

<li><a href="Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li>

<li><a href="Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul>

<ul><li><a href="Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li>

<li><a href="Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Cauchy's_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li>

<li><a href="Lagrange's_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul>

<ul><li><a href="Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li>

<li><a href="Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul>

<ul><li><a href="P-group" title="P-group"><i>p</i>-group</a></li>

<li><a href="Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul>

<ul><li><a href="Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul>

<ul><li><a href="Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<a href="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="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="Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="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="Integer" title="Integer">Integers</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 \mathbb {Z} }">

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<li><a href="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="Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <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 \mathbb {Z} }">

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

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</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" 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} }" loading="lazy"></span>)</li><li>SL(2, <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 \mathbb {Z} }">

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

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<ul><li><a href="Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li>

<li><a href="Lattice_(group)" title="Lattice (group)">Lattice</a></li>

<li><a href="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="Topological_group" title="Topological group">Topological</a> and <a href="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="Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li>

<li><a href="Circle_group" title="Circle group">Circle</a></li></ul>

<ul><li><a href="General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul>

<ul><li><a href="Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul>

<ul><li><a href="Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul>

<ul><li><a href="Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul>

<ul><li><a href="Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul>

<ul><li><a href="Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul>

<ul><li><a href="Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul>

<ul><li><a href="Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul>

<ul><li><a href="Lorentz_group" title="Lorentz group">Lorentz</a></li>

<li><a href="Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li>

<li><a href="Conformal_group" title="Conformal group">Conformal</a></li></ul>

<ul><li><a href="Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li>

<div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="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">

<ul><li><a href="Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul>

<ul><li><a href="Reductive_group" title="Reductive group">Reductive group</a></li></ul>

<ul><li><a href="Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul>

<ul><li><a href="Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>simple group</b> is a nontrivial <a href="Group_(mathematics)" title="Group (mathematics)">group</a> whose only <a href="Normal_subgroup" title="Normal subgroup">normal subgroups</a> are the <a href="Trivial_group" title="Trivial group">trivial group</a> and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding <a href="Quotient_group" title="Quotient group">quotient group</a>. This process can be repeated, and for <a href="Finite_group" title="Finite group">finite groups</a> one eventually arrives at uniquely determined simple groups, by the <a href="Jordan%E2%80%93H%C3%B6lder_theorem" class="mw-redirect" title="Jordan–Hölder theorem">Jordan–Hölder theorem</a>.

</p><p>The complete <a href="Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>, completed in 2004, is a major milestone in the history of mathematics.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Finite_simple_groups">Finite simple groups</h3></div>

<p>The <a href="Cyclic_group" title="Cyclic group">cyclic 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=(\mathbb {Z} /3\mathbb {Z} ,+)=\mathbb {Z} _{3}}">

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<annotation encoding="application/x-tex">{\displaystyle G=(\mathbb {Z} /3\mathbb {Z} ,+)=\mathbb {Z} _{3}}</annotation>

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</math></span><img src="./7a65e93f706d42beafad6ea3fda278b97b3b3e39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.705ex; height:2.843ex;" alt="{\displaystyle G=(\mathbb {Z} /3\mathbb {Z} ,+)=\mathbb {Z} _{3}}" loading="lazy"></span> of <a href="Congruence_class" class="mw-redirect" title="Congruence class">congruence classes</a> <a href="Modulo_operation" class="mw-redirect" title="Modulo operation">modulo</a> 3 (see <a href="Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>) is simple. If <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 H}">

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is a subgroup of this group, its <a href="Order_(group_theory)" title="Order (group theory)">order</a> (the number of elements) must be a <a href="Divisor" title="Divisor">divisor</a> of the order of <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}">

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

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</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> which is 3. Since 3 is prime, its only divisors are 1 and 3, so either <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 H}">

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

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

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</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>, 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 H}">

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

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is the trivial group. On the other hand, the group <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=(\mathbb {Z} /12\mathbb {Z} ,+)=\mathbb {Z} _{12}}">

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=(\mathbb {Z} /12\mathbb {Z} ,+)=\mathbb {Z} _{12}}</annotation>

</semantics>

</math></span><img src="./6a11575e4ab00645972569a7f08711e2bdf1c12e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.69ex; height:2.843ex;" alt="{\displaystyle G=(\mathbb {Z} /12\mathbb {Z} ,+)=\mathbb {Z} _{12}}" loading="lazy"></span> is not simple. The set <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 H}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an <a href="Abelian_group" title="Abelian group">abelian group</a> is normal. Similarly, the additive group of the <a href="Integer" title="Integer">integers</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 (\mathbb {Z} ,+)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} ,+)}</annotation>

</semantics>

</math></span><img src="./910eaae0a8267ccb04d4846f6a28f02ce6ab8ac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.202ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} ,+)}" loading="lazy"></span> is not simple; the set of even integers is a non-trivial proper normal subgroup.<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><p>One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of <a href="Prime_number" title="Prime number">prime</a> order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the <a href="Alternating_group" title="Alternating group">alternating 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 A_{5}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{5}}</annotation>

</semantics>

</math></span><img src="./6e213bbb69691c65e1391fe16cd79a0029471446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{5}}" loading="lazy"></span>.<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> The second smallest nonabelian simple group is the projective special linear group <a href="PSL(2%2C7)" title="PSL(2,7)">PSL(2,7)</a> of order 168, and every simple group of order 168 is isomorphic to PSL(2,7).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Infinite_simple_groups">Infinite simple groups</h3></div>

<p>The infinite alternating group <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_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./15b6dfe5968776343496f22a0a90c8406065def1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{\infty }}" loading="lazy"></span>, i.e. the group of even finitely supported permutations of the integers, is simple. This group can be written as the increasing union of the finite simple groups <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_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./730f6906700685b6d52f3958b1c2ae659d2d97d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}" loading="lazy"></span> with respect to standard embeddings <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_{n}\rightarrow A_{n+1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./29078c5b715c2990a2e5b9f112cea30c997d4fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.638ex; height:2.509ex;" alt="{\displaystyle A_{n}\rightarrow A_{n+1}}" loading="lazy"></span>. Another family of examples of infinite simple groups is given by <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 PSL_{n}(F)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PSL_{n}(F)}</annotation>

</semantics>

</math></span><img src="./60d429e6d5587016f44bb3e382ec45598dcf238f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.596ex; height:2.843ex;" alt="{\displaystyle PSL_{n}(F)}" loading="lazy"></span>, 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 F}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> is an infinite field and <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 n\geq 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation>

</semantics>

</math></span><img src="./e6bf67f9d06ca3af619657f8d20ee1322da77174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}" loading="lazy"></span>.

</p><p>It is much more difficult to construct <i>finitely generated</i> infinite simple groups. The first existence result is non-explicit; it is due to <a href="Graham_Higman" title="Graham Higman">Graham Higman</a> and consists of simple quotients of the <a href="Higman_group" title="Higman group">Higman group</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Explicit examples, which turn out to be finitely presented, include the infinite <a href="Thompson_groups" title="Thompson groups">Thompson groups</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 T}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> and <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 V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>. Finitely presented <a href="Torsion_(algebra)" title="Torsion (algebra)">torsion-free</a> infinite simple groups were constructed by Burger and Mozes.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>There is as yet no known classification for general (infinite) simple groups, and no such classification is expected. One reason for this is the existence of continuum-many <a href="Tarski_monster_group" title="Tarski monster group">Tarski monster groups</a> for every sufficiently-large prime characteristic, each simple and having only the cyclic group of that characteristic as its subgroups.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Finite_simple_groups_2">Finite simple groups</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="List_of_finite_simple_groups" title="List of finite simple groups">list of finite simple groups</a></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></div>

<p>The <a href="List_of_finite_simple_groups" title="List of finite simple groups">finite simple groups</a> are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way <a href="Prime_number" title="Prime number">prime numbers</a> are the basic building blocks of the <a href="Integer" title="Integer">integers</a>. This is expressed by the <a href="Jordan%E2%80%93H%C3%B6lder_theorem" class="mw-redirect" title="Jordan–Hölder theorem">Jordan–Hölder theorem</a> which states that any two <a href="Composition_series" title="Composition series">composition series</a> of a given group have the same length and the same factors, <a href="Up_to" title="Up to">up to</a> <a href="Permutation" title="Permutation">permutation</a> and <a href="Isomorphism" title="Isomorphism">isomorphism</a>. In a huge collaborative effort, the <a href="Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> was declared accomplished in 1983 by <a href="Daniel_Gorenstein" title="Daniel Gorenstein">Daniel Gorenstein</a>, though some problems surfaced (specifically in the classification of <a href="Quasithin_group" title="Quasithin group">quasithin groups</a>, which were plugged in 2004).

</p><p>Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:

</p>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

</semantics>

</math></span><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy"></span> – <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of prime order</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./730f6906700685b6d52f3958b1c2ae659d2d97d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}" loading="lazy"></span> – <a href="Alternating_group" title="Alternating group">alternating group</a> 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 n\geq 5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 5}</annotation>

</semantics>

</math></span><img src="./1a752e15bfe1dac8d617d014a77c275bfd4af0d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 5}" loading="lazy"></span>

<dl><dd>The alternating groups may be considered as groups of Lie type over the <a href="Field_with_one_element" title="Field with one element">field with one element</a>, which unites this family with the next, and thus all families of non-abelian finite simple groups may be considered to be of Lie type.</dd></dl></li>

<li>One of 16 families of <a href="Groups_of_Lie_type" class="mw-redirect" title="Groups of Lie type">groups of Lie type</a> or their derivatives

<dl><dd>The <a href="Tits_group" title="Tits group">Tits group</a> is generally considered of this form, though strictly speaking it is not of Lie type, but rather index 2 in a group of Lie type.</dd></dl></li>

<li>One of 26 exceptions, the <a href="Sporadic_group" title="Sporadic group">sporadic groups</a>, of which 20 are subgroups or <a href="Subquotient" title="Subquotient">subquotients</a> of the <a href="Monster_group" title="Monster group">monster group</a> and are referred to as the "Happy Family", while the remaining 6 are referred to as <a href="Pariah_group" title="Pariah group">pariahs</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Structure_of_finite_simple_groups">Structure of finite simple groups</h2></div>

<p>The famous <a href="Feit%E2%80%93Thompson_theorem" title="Feit–Thompson theorem">theorem</a> of <a href="Walter_Feit" title="Walter Feit">Feit</a> and <a href="John_G._Thompson" title="John G. Thompson">Thompson</a> states that every group of odd order is <a href="Solvable_group" title="Solvable group">solvable</a>. Therefore, every finite simple group has even order unless it is cyclic of prime order.

</p><p>The <a href="Schreier_conjecture" title="Schreier conjecture">Schreier conjecture</a> asserts that the group of <a href="Outer_automorphism" class="mw-redirect" title="Outer automorphism">outer automorphisms</a> of every finite simple group is solvable. This can be proved using the <a href="Classification_theorem" title="Classification theorem">classification theorem</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="History_for_finite_simple_groups">History for finite simple groups</h2></div>

<p>There are two threads in the history of finite simple groups – the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the Monster in 1981; and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 (when victory was initially declared), but was only generally agreed to be finished in 2004. By 2018, its publication was envisioned as a series of 12 <a href="Monograph" title="Monograph">monographs</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> the tenth of which was published in 2023.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> See (<a href="#CITEREFSilvestri1979">Silvestri 1979</a>) for 19th century history of simple groups.

</p>

<div class="mw-heading mw-heading3"><h3 id="Construction">Construction</h3></div>

<p>Simple groups have been studied at least since early <a href="Galois_theory" title="Galois theory">Galois theory</a>, where <a href="%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> realized that the fact that the <a href="Alternating_group" title="Alternating group">alternating groups</a> on five or more points are simple (and hence not solvable), which he proved in 1831, was the reason that one could not solve the quintic in radicals. Galois also constructed the <a href="Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear group</a> of a plane over a prime finite field, <span class="nowrap">PSL(2,<i>p</i>)</span>, and remarked that they were simple for <i>p</i> not 2 or 3. This is contained in his last letter to Chevalier,<sup id="cite_ref-chevalier-letter_10-0" class="reference"><a href="#cite_note-chevalier-letter-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and are the next example of finite simple groups.<sup id="cite_ref-raw_11-0" class="reference"><a href="#cite_note-raw-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p><p>The next discoveries were by <a href="Camille_Jordan" title="Camille Jordan">Camille Jordan</a> in 1870.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Jordan had found 4 families of simple matrix groups over <a href="Finite_field" title="Finite field">finite fields</a> of prime order, which are now known as the <a href="Classical_group" title="Classical group">classical groups</a>.

</p><p>At about the same time, it was shown that a family of five groups, called the <a href="Mathieu_group" title="Mathieu group">Mathieu groups</a> and first described by <a href="%C3%89mile_L%C3%A9onard_Mathieu" title="Émile Léonard Mathieu">Émile Léonard Mathieu</a> in 1861 and 1873, were also simple. Since these five groups were constructed by methods which did not yield infinitely many possibilities, they were called "<a href="Sporadic_group" title="Sporadic group">sporadic</a>" by <a href="William_Burnside" title="William Burnside">William Burnside</a> in his 1897 textbook.

</p><p>Later Jordan's results on classical groups were generalized to arbitrary finite fields by <a href="Leonard_Dickson" class="mw-redirect" title="Leonard Dickson">Leonard Dickson</a>, following the classification of <a href="Complex_simple_Lie_algebra" class="mw-redirect" title="Complex simple Lie algebra">complex simple Lie algebras</a> by <a href="Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a>. Dickson also constructed exception groups of type G<sub>2</sub> and <a href="E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a> as well, but not of types F<sub>4</sub>, E<sub>7</sub>, or E<sub>8</sub> (<a href="#CITEREFWilson2009">Wilson 2009</a>, p. 2). In the 1950s the work on groups of Lie type was continued, with <a href="Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a> giving a uniform construction of the classical groups and the groups of exceptional type in a 1955 paper. This omitted certain known groups (the projective unitary groups), which were obtained by "twisting" the Chevalley construction. The remaining groups of Lie type were produced by Steinberg, Tits, and Herzig (who produced <sup>3</sup><i>D</i><sub>4</sub>(<i>q</i>) and <sup>2</sup><i>E</i><sub>6</sub>(<i>q</i>)) and by Suzuki and Ree (the <a href="Suzuki%E2%80%93Ree_group" class="mw-redirect" title="Suzuki–Ree group">Suzuki–Ree groups</a>).

</p><p>These groups (the groups of Lie type, together with the cyclic groups, alternating groups, and the five exceptional Mathieu groups) were believed to be a complete list, but after a lull of almost a century since the work of Mathieu, in 1964 the first <a href="Janko_group" title="Janko group">Janko group</a> was discovered, and the remaining 20 sporadic groups were discovered or conjectured in 1965–1975, culminating in 1981, when <a href="Robert_Griess" title="Robert Griess">Robert Griess</a> announced that he had constructed <a href="Bernd_Fischer_(mathematician)" title="Bernd Fischer (mathematician)">Bernd Fischer</a>'s "<a href="Monster_group" title="Monster group">Monster group</a>". The Monster is the largest sporadic simple group having order of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. The Monster has a faithful 196,883-dimensional representation in the 196,884-dimensional <a href="Griess_algebra" title="Griess algebra">Griess algebra</a>, meaning that each element of the Monster can be expressed as a 196,883 by 196,883 matrix.

</p>

<div class="mw-heading mw-heading3"><h3 id="Classification_2">Classification</h3></div>

<p>The full classification is generally accepted as beginning with the <a href="Feit%E2%80%93Thompson_theorem" title="Feit–Thompson theorem">Feit–Thompson theorem</a> of 1962–1963 and being completed in 2004.

</p><p>Soon after the construction of the Monster in 1981, a proof, totaling more than 10,000 pages, was supplied in 1983 by Daniel Gorenstein, that claimed to successfully <a href="List_of_finite_simple_groups" title="List of finite simple groups">list all finite simple groups</a>. This was premature, as gaps were later discovered in the classification of <a href="Quasithin_group" title="Quasithin group">quasithin groups</a>. The gaps were filled in 2004 by a 1300 page classification of quasithin groups and the proof is now generally accepted as complete.

</p>

<div class="mw-heading mw-heading2"><h2 id="Tests_for_nonsimplicity">Tests for nonsimplicity</h2></div>

<p><i><a href="Sylow_theorems#Example_applications" title="Sylow theorems">Sylow's test</a></i>: Let <i>n</i> be a positive integer that is not prime, and let <i>p</i> be a prime divisor of <i>n</i>. If 1 is the only divisor of <i>n</i> that is congruent to 1 modulo <i>p</i>, then there does not exist a simple group of order <i>n</i>.

</p><p>Proof: If <i>n</i> is a prime-power, then a group of order <i>n</i> has a nontrivial <a href="Center_(group_theory)" title="Center (group theory)">center</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and, therefore, is not simple. If <i>n</i> is not a prime power, then every Sylow subgroup is proper, and, by <a href="Sylow_theorems" title="Sylow theorems">Sylow's Third Theorem</a>, we know that the number of Sylow <i>p</i>-subgroups of a group of order <i>n</i> is equal to 1 modulo <i>p</i> and divides <i>n</i>. Since 1 is the only such number, the Sylow <i>p</i>-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.

</p><p><i>Burnside</i>: A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from <a href="Burnside's_theorem" title="Burnside's theorem">Burnside's theorem</a>.

</p>

<ul><li><a href="Almost_simple_group" title="Almost simple group">Almost simple group</a></li>

<li><a href="Characteristically_simple_group" title="Characteristically simple group">Characteristically simple group</a></li>

<li><a href="Quasisimple_group" title="Quasisimple group">Quasisimple group</a></li>

<li><a href="Semisimple_group" class="mw-redirect" title="Semisimple group">Semisimple group</a></li>

<li><a href="List_of_finite_simple_groups" title="List of finite simple groups">List of finite simple groups</a></li></ul>

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

<div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Knapp (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KVeXG163BggC&pg=PA170&dq=%22Z+is+not+simple%2C+having+the+nontrivial+subgroup+2Z%22">p. 170</a></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Rotman (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lYrsiaHSHKcC&pg=PA226&dq=%22simple+groups+of+order+60+are+isomorphic%22">p. 226</a></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Rotman (1995), p. 281</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Smith & Tabachnikova (2000), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DD0TW28WjfQC&pg=PA144&dq=%22any+two+simple+groups+of+order+168+are+isomorphic%22">p. 144</a></span>

</li>

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</style><cite id="CITEREFHigman1951" class="citation cs2"><a href="Graham_Higman" title="Graham Higman">Higman, Graham</a> (1951), "A finitely generated infinite simple group", <i>Journal of the London Mathematical Society</i>, Second Series, <b>26</b> (1): <span class="nowrap">61–</span>64, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjlms%2Fs1-26.1.59">10.1112/jlms/s1-26.1.59</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0024-6107">0024-6107</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0038348">0038348</a></cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFBurgerMozes2000" class="citation journal cs1">Burger, M.; Mozes, S. (2000). "Lattices in product of trees". <i>Publ. Math. IHÉS</i>. <b>92</b>: <span class="nowrap">151–</span>194. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02698916">10.1007/bf02698916</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55003601">55003601</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFOtal2004" class="citation cs2">Otal, Javier (2004), <a rel="nofollow" class="external text" href="https://www.raczar.es/webracz/ImageServlet?mod=publicaciones&subMod=monografias&car=monografia26&archivo=089Otal.pdf">"The Classification of the Finite Simple Groups: An Overview"</a> <span class="cs1-format">(PDF)</span>, in Boya, L. J. (ed.), <i>Problemas del Milenio</i>, Monografías de la Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza, vol. 26, Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFSolomon2018" class="citation cs2">Solomon, Ronald (2018), <a rel="nofollow" class="external text" href="https://www.ams.org/journals/notices/201806/rnoti-p646.pdf">"The classification of finite simple groups: a progress report"</a> <span class="cs1-format">(PDF)</span>, <i>Notices of the American Mathematical Society</i>, <b>65</b> (6): <span class="nowrap">646–</span>651, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti1689">10.1090/noti1689</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3792856">3792856</a></cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFCapdeboscqGorensteinLyonsSolomon2023" class="citation cs2">Capdeboscq, Inna; Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald (2023), <i>The classification of the finite simple groups, Number 10. Part V. Chapters 9–17. Theorem <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 C_{6}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{6}}</annotation>

</semantics>

</math></span><img src="./17692472a5b1f5de8b8b27603154198c16b61fbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{6}}" loading="lazy"></span> and Theorem <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 C_{4}^{\ast }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{4}^{\ast }}</annotation>

</semantics>

</math></span><img src="./75f29b6eabbb66dd4915238ee7cc4f631f6444ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{4}^{\ast }}" loading="lazy"></span>, Case A</i>, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4704-7553-6</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=4656413">4656413</a></cite></span>

</li>

<li id="cite_note-chevalier-letter-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-chevalier-letter_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGalois1846" class="citation cs2">Galois, Évariste (1846), <a rel="nofollow" class="external text" href="http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-16390&I=416&M=tdm">"Lettre de Galois à M. Auguste Chevalier"</a>, <i><a href="Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es" title="Journal de Mathématiques Pures et Appliquées">Journal de Mathématiques Pures et Appliquées</a></i>, <b>XI</b>: <span class="nowrap">408–</span>415<span class="reference-accessdate">, retrieved <span class="nowrap">2009-02-04</span></span>, PSL(2,<i>p</i>) and simplicity are discussed on p. 411; exceptional action on 5, 7, or 11 points is discussed on pp. 411–412; GL(<i>ν</i>,<i>p</i>) is discussed on p. 410</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{citation}}</code>: CS1 maint: postscript (link)</span></span>

</li>

<li id="cite_note-raw-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-raw_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilson2006" class="citation cs2"><a href="Robert_Arnott_Wilson" title="Robert Arnott Wilson">Wilson, Robert</a> (October 31, 2006), <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps">"Chapter 1: Introduction"</a>, <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~raw/fsgs.html"><i>The finite simple groups</i></a></cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFJordan1870" class="citation cs2"><a href="Camille_Jordan" title="Camille Jordan">Jordan, Camille</a> (1870), <i><a href="List_of_important_publications_in_mathematics" class="mw-redirect" title="List of important publications in mathematics">Traité des substitutions et des équations algébriques</a></i></cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">See the proof in <a href="P-group" title="P-group"><i>p</i>-group</a>, for instance.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Textbooks">Textbooks</h3></div>

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<ul><li><cite id="CITEREFWilson2009" class="citation cs2"><a href="Robert_Arnott_Wilson" title="Robert Arnott Wilson">Wilson, Robert A.</a> (2009), <i>The finite simple groups</i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a> 251, vol. 251, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-84800-988-2">10.1007/978-1-84800-988-2</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84800-987-5</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1203.20012">1203.20012</a></cite>; <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~raw/fsgs.html">2007 preprint</a>.</li>

<li><cite id="CITEREFBurnside1897" class="citation cs2"><a href="William_Burnside" title="William Burnside">Burnside, William</a> (1897), <i>Theory of groups of finite order</i>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a></cite></li></ul>

</div>

<ul><li><cite id="CITEREFKnapp2006" class="citation cs2">Knapp, Anthony W. (2006), <i>Basic algebra</i>, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-3248-9</bdi></cite></li>

<li><cite id="CITEREFRotman1995" class="citation cs2">Rotman, Joseph J. (1995), <i>An introduction to the theory of groups</i>, Graduate texts in mathematics, vol. 148, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94285-8</bdi></cite></li>

<li><cite id="CITEREFSmithTabachnikova2000" class="citation cs2">Smith, Geoff; Tabachnikova, Olga (2000), <i>Topics in group theory</i>, Springer undergraduate mathematics series (2 ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-85233-235-8</bdi></cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Papers">Papers</h3></div>

<ul><li><cite id="CITEREFSilvestri1979" class="citation cs2">Silvestri, R. (September 1979), "Simple groups of finite order in the nineteenth century", <i>Archive for History of Exact Sciences</i>, <b>20</b> (<span class="nowrap">3–</span>4): <span class="nowrap">313–</span>356, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00327738">10.1007/BF00327738</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120444304">120444304</a></cite></li></ul>

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