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Conjugacy class

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In <a href="Mathematics" title="Mathematics">mathematics</a>, especially <a href="Group_theory" title="Group theory">group theory</a>, two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> of a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> are conjugate if there is an element <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><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in the group such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=gag^{-1}.}">

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

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</math><img src="./f3a0617f1a4881d37dc80a6a6a9dc0a5256b8803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:3.009ex;" alt="{\displaystyle b=gag^{-1}.}" loading="lazy"> This is an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> whose <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> are called conjugacy classes. In other words, each conjugacy class is closed under <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=gag^{-1}}">

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

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</math><img src="./bcd1163569726bbaf3cd0726b1134705d6a06b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.893ex; height:3.009ex;" alt="{\displaystyle b=gag^{-1}}" loading="lazy"> for all elements <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><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in the group.

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of <a href="Non-abelian_group" title="Non-abelian group">non-abelian groups</a> is fundamental for the study of their structure.<a href="#cite_note-dummit-1">[1]</a><a href="#cite_note-2">[2]</a> For an <a href="Abelian_group" title="Abelian group">abelian group</a>, each conjugacy class is a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> containing one element (<a href="Singleton_(mathematics)" title="Singleton (mathematics)">singleton set</a>).

<a href="Function_(mathematics)" title="Function (mathematics)">Functions</a> that are constant for members of the same conjugacy class are called <a href="Class_function" title="Class function">class functions</a>.

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

Let <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><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"> be a group. Two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in G}">

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

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</math><img src="./a4b3751cbf424027e1e00e22d191a2d465403c79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.929ex; height:2.509ex;" alt="{\displaystyle a,b\in G}" loading="lazy"> are conjugate if there exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

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

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</math><img src="./b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gag^{-1}=b,}">

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

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</math><img src="./2f3a3509d1a9e458a99b84bc075f77ee797a7d1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:3.009ex;" alt="{\displaystyle gag^{-1}=b,}" loading="lazy"> in which case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is called a conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is called a conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b.}">

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

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</math><img src="./aef051eb30c89e5493d672f6479566c673b0890a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.644ex; height:2.176ex;" alt="{\displaystyle b.}" loading="lazy">

In the case of the <a href="General_linear_group" title="General linear group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n)}">

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

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</math><img src="./7033760b4d9a933782f9809f79b5c17e952453f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.481ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n)}" loading="lazy"> of <a href="Invertible_matrices" class="mw-redirect" title="Invertible matrices">invertible matrices</a>, the conjugacy relation is called <a href="Matrix_similarity" title="Matrix similarity">matrix similarity</a>.

It can be easily shown that conjugacy is an equivalence relation and therefore partitions <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><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"> into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} (a)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} (a)}</annotation>

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</math><img src="./f238ba8841819097c4acb1c98a0c83232d73db02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.364ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} (a)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} (b)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} (b)}</annotation>

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</math><img src="./f673081fed9777c56f76e3b5ae9fbda612b37655.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.132ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} (b)}" loading="lazy"> are equal <a href="If_and_only_if" title="If and only if">if and only if</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are conjugate, and <a href="Disjoint_sets" title="Disjoint sets">disjoint</a> otherwise.) The equivalence class that contains the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in G}">

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

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</math><img src="./fe9f5ea7aea0b7a62b07eae139e7a5038ea5a120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle a\in G}" loading="lazy"> is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} (a)=\left\{gag^{-1}:g\in G\right\}}</annotation>

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and is called the conjugacy class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}">

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

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</math><img src="./6b803da9c45c1186883bde55107e9ccb102c92c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}" loading="lazy"> The <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>class number of <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><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"> is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same <a href="Order_(group_theory)" title="Order (group theory)">order</a>.

Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1. In some cases, conjugacy classes can be described in a uniform way; for example, in the <a href="Symmetric_group" title="Symmetric group">symmetric group</a> they can be described by <a href="Permutation#Cycle_type" title="Permutation">cycle type</a>.

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

The symmetric group <a href="Dihedral_group_of_order_6" title="Dihedral group of order 6"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{3},}">

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<annotation encoding="application/x-tex">{\displaystyle S_{3},}</annotation>

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</math><img src="./791fcc238d3688eddfd96d84556ede2bc7b93a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.126ex; height:2.509ex;" alt="{\displaystyle S_{3},}" loading="lazy"></a> consisting of the 6 <a href="Permutation" title="Permutation">permutations</a> of three elements, has three conjugacy classes:

<ol><li>No change <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (abc\to abc)}">

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<annotation encoding="application/x-tex">{\displaystyle (abc\to abc)}</annotation>

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</math><img src="./77c2ec6ab88ff1615b2a66a806548caa5bdc894f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.843ex;" alt="{\displaystyle (abc\to abc)}" loading="lazy">. The single member has order 1.</li>

<li><a href="Cyclic_permutation#Transpositions" title="Cyclic permutation">Transposing</a> two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (abc\to acb,abc\to bac,abc\to cba)}">

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<annotation encoding="application/x-tex">{\displaystyle (abc\to acb,abc\to bac,abc\to cba)}</annotation>

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</math><img src="./1854ce0816951520422274a263943edeb9df89c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.125ex; height:2.843ex;" alt="{\displaystyle (abc\to acb,abc\to bac,abc\to cba)}" loading="lazy">. The 3 members all have order 2.</li>

<li>A <a href="Cyclic_permutation" title="Cyclic permutation">cyclic permutation</a> of all three <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (abc\to bca,abc\to cab)}">

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<annotation encoding="application/x-tex">{\displaystyle (abc\to bca,abc\to cab)}</annotation>

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</math><img src="./e9d472096baeb51c2152422cbcf9928d242d4de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.008ex; height:2.843ex;" alt="{\displaystyle (abc\to bca,abc\to cab)}" loading="lazy">. The 2 members both have order 3.</li></ol>

These three classes also correspond to the classification of the <a href="Isometry_group" title="Isometry group">isometries</a> of an <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>.

The symmetric group <a href="https://en.wikiversity.org/wiki/Symmetric_group_S4" class="extiw external" title="v:Symmetric group S4"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{4},}">

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<annotation encoding="application/x-tex">{\displaystyle S_{4},}</annotation>

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</math><img src="./95c0ad1b3affc17414a69e4ebe37b107fa765bbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.126ex; height:2.509ex;" alt="{\displaystyle S_{4},}" loading="lazy"></a> consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description, <a href="Permutation#Cycle_type" title="Permutation">cycle type</a>, member order, and members:

<ol><li>No change. Cycle type = [14]. Order = 1. Members = { (1, 2, 3, 4) }. The single row containing this conjugacy class is shown as a row of black circles in the adjacent table.</li>

<li>Interchanging two (other two remain unchanged). Cycle type = [1221]. Order = 2. Members = { (1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (4, 2, 3, 1), (3, 2, 1, 4), (2, 1, 3, 4) }). The 6 rows containing this conjugacy class are highlighted in green in the adjacent table.</li>

<li>A cyclic permutation of three (other one remains unchanged). Cycle type = [1131]. Order = 3. Members = { (1, 3, 4, 2), (1, 4, 2, 3), (3, 2, 4, 1), (4, 2, 1, 3), (4, 1, 3, 2), (2, 4, 3, 1), (3, 1, 2, 4), (2, 3, 1, 4) }). The 8 rows containing this conjugacy class are shown with normal print (no boldface or color highlighting) in the adjacent table.</li>

<li>A cyclic permutation of all four. Cycle type = [41]. Order = 4. Members = { (2, 3, 4, 1), (2, 4, 1, 3), (3, 1, 4, 2), (3, 4, 2, 1), (4, 1, 2, 3), (4, 3, 1, 2) }). The 6 rows containing this conjugacy class are highlighted in orange in the adjacent table.</li>

<li>Interchanging two, and also the other two. Cycle type = [22]. Order = 2. Members = { (2, 1, 4, 3), (4, 3, 2, 1), (3, 4, 1, 2) }). The 3 rows containing this conjugacy class are shown with boldface entries in the adjacent table.</li></ol>

In general, the number of conjugacy classes in the symmetric group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}">

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

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</math><img src="./9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}" loading="lazy"> is equal to the number of <a href="Integer_partition" title="Integer partition">integer partitions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}">

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

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</math><img src="./e59df02a9f67a5da3c220f1244c99a46cc4eb1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}" loading="lazy"> This is because each conjugacy class corresponds to exactly one partition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,n\}}">

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

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</math><img src="./4ebfec86b3f22a18f086275390917d5aaa2d8c22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,n\}}" loading="lazy"> into <a href="Cycle_notation" class="mw-redirect" title="Cycle notation">cycles</a>, up to permutation of the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,n\}.}">

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

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</math><img src="./ab0db9d1eaa9b943133923479792d31d0d995c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.904ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,n\}.}" loading="lazy">

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

<ul><li>The <a href="Identity_element" title="Identity element">identity element</a> is always the only element in its class, that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} (e)=\{e\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} (e)=\{e\}.}</annotation>

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</math><img src="./d3bbb88244e45b057d09e99fa2696121bcba658c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.371ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} (e)=\{e\}.}" loading="lazy"></li>

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

</semantics>

</math><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"> is <a href="Abelian_group" title="Abelian group">abelian</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gag^{-1}=a}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f72b562b3a843e993ce704b854e57d78bc779361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.125ex; height:3.009ex;" alt="{\displaystyle gag^{-1}=a}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,g\in G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./61f7ddb0330abeeb97e1ded6830c491bcf705f5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.047ex; height:2.509ex;" alt="{\displaystyle a,g\in G}" loading="lazy">, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} (a)=\{a\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} (a)=\{a\}}</annotation>

</semantics>

</math><img src="./3901449ea6f2f19e984f5100318316b109f94601.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.017ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} (a)=\{a\}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fe9f5ea7aea0b7a62b07eae139e7a5038ea5a120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle a\in G}" loading="lazy"> (and the converse is also true: if all conjugacy classes are singletons then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is abelian).</li>

<li>If two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a4b3751cbf424027e1e00e22d191a2d465403c79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.929ex; height:2.509ex;" alt="{\displaystyle a,b\in G}" loading="lazy"> belong to the same conjugacy class (that is, if they are conjugate), then they have the same <a href="Order_(group_theory)" title="Order (group theory)">order</a>. More generally, every statement about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> can be translated into a statement about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=gag^{-1},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a9eb338105e5faccd021323f21dcdbc7b2403359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.54ex; height:3.009ex;" alt="{\displaystyle b=gag^{-1},}" loading="lazy"> because the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x)=gxg^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (x)=gxg^{-1}}</annotation>

</semantics>

</math><img src="./de96b66c9dd0d2f7a36b732409d55113dc3cc808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.654ex; height:3.176ex;" alt="{\displaystyle \varphi (x)=gxg^{-1}}" loading="lazy"> is an <a href="Group_isomorphism#Automorphisms" title="Group isomorphism">automorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> called an <a href="Inner_automorphism" title="Inner automorphism">inner automorphism</a>. See the next property for an example.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are conjugate, then so are their powers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{k}}</annotation>

</semantics>

</math><img src="./79e30fb4fe7c63445f6f0f433e87ae65d87ff55f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.319ex; height:2.676ex;" alt="{\displaystyle a^{k}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{k}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{k}.}</annotation>

</semantics>

</math><img src="./1129f71d07ec04a8b05cf85009704a9216091425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.733ex; height:2.676ex;" alt="{\displaystyle b^{k}.}" loading="lazy"> (Proof: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=gbg^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fba6b377aa10a9af88cca3ac84f850ab3e1f4554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.893ex; height:3.009ex;" alt="{\displaystyle a=gbg^{-1}}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{k}=\left(gbg^{-1}\right)\left(gbg^{-1}\right)\cdots \left(gbg^{-1}\right)=gb^{k}g^{-1}.}">

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{k}=\left(gbg^{-1}\right)\left(gbg^{-1}\right)\cdots \left(gbg^{-1}\right)=gb^{k}g^{-1}.}</annotation>

</semantics>

</math><img src="./078a3571e7e1d137eb20ea35a8895e654d7288a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.783ex; height:3.343ex;" alt="{\displaystyle a^{k}=\left(gbg^{-1}\right)\left(gbg^{-1}\right)\cdots \left(gbg^{-1}\right)=gb^{k}g^{-1}.}" loading="lazy">) Thus taking kth powers gives a map on conjugacy classes, and one may consider which conjugacy classes are in its preimage. For example, in the symmetric group, the square of an element of type (3)(2) (a 3-cycle and a 2-cycle) is an element of type (3), therefore one of the power-up classes of (3) is the class (3)(2) (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is a power-up class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{k}}</annotation>

</semantics>

</math><img src="./79e30fb4fe7c63445f6f0f433e87ae65d87ff55f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.319ex; height:2.676ex;" alt="{\displaystyle a^{k}}" loading="lazy">).</li>

<li>An element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fe9f5ea7aea0b7a62b07eae139e7a5038ea5a120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle a\in G}" loading="lazy"> lies in the <a href="Center_of_a_group" class="mw-redirect" title="Center of a group">center</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Z} (G)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b036445264ec4b243b84f99518ac7c2a2828724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.056ex; height:2.843ex;" alt="{\displaystyle \operatorname {Z} (G)}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> if and only if its conjugacy class has only one element, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> itself. More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {C} _{G}(a)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {C} _{G}(a)}</annotation>

</semantics>

</math><img src="./9ab42af07c080785a663fc66ad4f94a169da46db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.241ex; height:2.843ex;" alt="{\displaystyle \operatorname {C} _{G}(a)}" loading="lazy"> denotes the <a href="Centralizer" class="mw-redirect" title="Centralizer">centralizer</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./76db749e92a69b27e81b370b6ec3a2a199420ee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.544ex; height:2.509ex;" alt="{\displaystyle a\in G,}" loading="lazy"> i.e., the <a href="Subgroup" title="Subgroup">subgroup</a> consisting of all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ga=ag,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ga=ag,}</annotation>

</semantics>

</math><img src="./a2687b8192123302ef935206d64d635c8c39275b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.437ex; height:2.009ex;" alt="{\displaystyle ga=ag,}" loading="lazy"> then the <a href="Index_of_a_subgroup" title="Index of a subgroup">index</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[G:\operatorname {C} _{G}(a)\right]}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[G:\operatorname {C} _{G}(a)\right]}</annotation>

</semantics>

</math><img src="./c54e724a6727915e7aad698e91d3b239bb663662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.299ex; height:2.843ex;" alt="{\displaystyle \left[G:\operatorname {C} _{G}(a)\right]}" loading="lazy"> is equal to the number of elements in the conjugacy class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> (by the <a href="Orbit-stabilizer_theorem" class="mw-redirect" title="Orbit-stabilizer theorem">orbit-stabilizer theorem</a>).</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2909dd9d8d53e51c7b767dbc347b6507e4a9f578.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.814ex; height:2.509ex;" alt="{\displaystyle \sigma \in S_{n}}" loading="lazy"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2},\ldots ,m_{s}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2},\ldots ,m_{s}}</annotation>

</semantics>

</math><img src="./67bfb5008c551242fab5252b43f23285465a00d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.445ex; height:2.009ex;" alt="{\displaystyle m_{1},m_{2},\ldots ,m_{s}}" loading="lazy"> be the distinct integers which appear as lengths of cycles in the cycle type of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> (including 1-cycles). Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f29138ed3ad54ffce527daccadc49c520459b0b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}" loading="lazy"> be the number of cycles of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./95ec8e804f69706d3f5ad235f4f983220c8df7c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2,\ldots ,s}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./648830d05d88abf1ad10a9223be2b6ea6057e853.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.509ex;" alt="{\displaystyle i=1,2,\ldots ,s}" loading="lazy"> (so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum \limits _{i=1}^{s}k_{i}m_{i}=n}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum \limits _{i=1}^{s}k_{i}m_{i}=n}</annotation>

</semantics>

</math><img src="./3c95bf10e5ac23230677c2039264ebd3c960cdd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.086ex; height:6.843ex;" alt="{\displaystyle \sum \limits _{i=1}^{s}k_{i}m_{i}=n}" loading="lazy">). Then the number of conjugates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is:<a href="#cite_note-dummit-1">[1]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n!}{\left(k_{1}!m_{1}^{k_{1}}\right)\left(k_{2}!m_{2}^{k_{2}}\right)\cdots \left(k_{s}!m_{s}^{k_{s}}\right)}}.}">

<mfrac>

<mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {n!}{\left(k_{1}!m_{1}^{k_{1}}\right)\left(k_{2}!m_{2}^{k_{2}}\right)\cdots \left(k_{s}!m_{s}^{k_{s}}\right)}}.}</annotation>

</semantics>

</math></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Conjugacy_as_group_action">Conjugacy as group action</h2></div>

For any two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,x\in G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65f4fd6d4f13fd4f37f406ac01d3f32ad72af200.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.509ex;" alt="{\displaystyle g,x\in G,}" loading="lazy"> let

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\cdot x:=gxg^{-1}.}</annotation>

</semantics>

</math>

This defines a <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy"> The <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">orbits</a> of this action are the conjugacy classes, and the <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">stabilizer</a> of a given element is the element's <a href="Centralizer" class="mw-redirect" title="Centralizer">centralizer</a>.<a href="#cite_note-Grillet-2007-p56-3">[3]</a>

Similarly, we can define a group action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> on the set of all <a href="Subset" title="Subset">subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> by writing

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\cdot S:=gSg^{-1},}</annotation>

</semantics>

</math>

or on the set of the subgroups of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Conjugacy_class_equation">Conjugacy class equation</h2></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is a <a href="Finite_group" title="Finite group">finite group</a>, then for any group element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f059f053fcf9f421b7c74362cf3bd5ed024e19d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}" loading="lazy"> the elements in the conjugacy class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> are in one-to-one correspondence with <a href="Coset" title="Coset">cosets</a> of the <a href="Centralizer" class="mw-redirect" title="Centralizer">centralizer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {C} _{G}(a).}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {C} _{G}(a).}</annotation>

</semantics>

</math><img src="./a898a3f98280c6ae4ff87c9a7c070886b22aecd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.888ex; height:2.843ex;" alt="{\displaystyle \operatorname {C} _{G}(a).}" loading="lazy"> This can be seen by observing that any two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> belonging to the same coset (and hence, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=cz}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7b774a414c1c6ceb2cf0e2ebccbcf63445d98643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.191ex; height:2.176ex;" alt="{\displaystyle b=cz}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> in the centralizer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {C} _{G}(a)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {C} _{G}(a)}</annotation>

</semantics>

</math><img src="./9ab42af07c080785a663fc66ad4f94a169da46db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.241ex; height:2.843ex;" alt="{\displaystyle \operatorname {C} _{G}(a)}" loading="lazy">) give rise to the same element when conjugating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle bab^{-1}=cza(cz)^{-1}=czaz^{-1}c^{-1}=cazz^{-1}c^{-1}=cac^{-1}.}</annotation>

</semantics>

</math>

That can also be seen from the <a href="Orbit-stabilizer_theorem" class="mw-redirect" title="Orbit-stabilizer theorem">orbit-stabilizer theorem</a>, when considering the group as acting on itself through conjugation, so that orbits are conjugacy classes and stabilizer subgroups are centralizers. The converse holds as well.

Thus the number of elements in the conjugacy class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is the <a href="Index_of_a_subgroup" title="Index of a subgroup">index</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[G:\operatorname {C} _{G}(a)\right]}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[G:\operatorname {C} _{G}(a)\right]}</annotation>

</semantics>

</math><img src="./c54e724a6727915e7aad698e91d3b239bb663662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.299ex; height:2.843ex;" alt="{\displaystyle \left[G:\operatorname {C} _{G}(a)\right]}" loading="lazy"> of the centralizer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {C} _{G}(a)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {C} _{G}(a)}</annotation>

</semantics>

</math><img src="./9ab42af07c080785a663fc66ad4f94a169da46db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.241ex; height:2.843ex;" alt="{\displaystyle \operatorname {C} _{G}(a)}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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">; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> from every conjugacy class, we infer from the disjointness of the conjugacy classes that

</mrow>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G|=\sum _{i}\left[G:\operatorname {C} _{G}(x_{i})\right],}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {C} _{G}(x_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {C} _{G}(x_{i})}</annotation>

</semantics>

</math><img src="./d1f8a98454de089131c433fe5fe255d61107b638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.141ex; height:2.843ex;" alt="{\displaystyle \operatorname {C} _{G}(x_{i})}" loading="lazy"> is the centralizer of the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ca686849ad350b47ebab4a394b66d84d5fa942bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.009ex;" alt="{\displaystyle x_{i}.}" loading="lazy"> Observing that each element of the center <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Z} (G)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b036445264ec4b243b84f99518ac7c2a2828724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.056ex; height:2.843ex;" alt="{\displaystyle \operatorname {Z} (G)}" loading="lazy"> forms a conjugacy class containing just itself gives rise to the class equation:<a href="#cite_note-4">[4]</a>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |G|=|{\operatorname {Z} (G)}|+\sum _{i}\left[G:\operatorname {C} _{G}(x_{i})\right],}</annotation>

</semantics>

</math>

where the sum is over a representative element from each conjugacy class that is not in the center.

Knowledge of the divisors of the group order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |G|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8258bc41edeb87bfbc8cba8367f29838c0eddc1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.12ex; height:2.843ex;" alt="{\displaystyle |G|}" loading="lazy"> can often be used to gain information about the order of the center or of the conjugacy classes.

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

Consider a finite <a href="P-group" title="P-group"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> (that is, a group with order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5984ca9fcd8f7b6424914768523ee36ebd6780b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.124ex; height:2.676ex;" alt="{\displaystyle p^{n},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a <a href="Prime_number" title="Prime number">prime number</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>

</semantics>

</math><img src="./27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy">). We are going to prove that every finite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-group has a non-<a href="Trivial_(mathematics)" class="mw-redirect" title="Trivial (mathematics)">trivial</a> center.

Since the order of any conjugacy class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> must divide the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> it follows that each conjugacy class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7bd0312f590cc5a400008938f3cc304d42ad3986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.731ex; height:2.509ex;" alt="{\displaystyle H_{i}}" loading="lazy"> that is not in the center also has order some power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k_{i}},}">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{k_{i}},}</annotation>

</semantics>

</math><img src="./581f2bd7ef08557c0a5ddd8410d22b4cc2a3e77c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.619ex; height:3.009ex;" alt="{\displaystyle p^{k_{i}},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<k_{i}<n.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<k_{i}<n.}</annotation>

</semantics>

</math><img src="./609b3a0689e2a9ecfcd6600405e715c12b17b893.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.412ex; height:2.509ex;" alt="{\displaystyle 0<k_{i}<n.}" loading="lazy"> But then the class equation requires that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |G|=p^{n}=|{\operatorname {Z} (G)}|+\sum _{i}p^{k_{i}}.}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle |G|=p^{n}=|{\operatorname {Z} (G)}|+\sum _{i}p^{k_{i}}.}</annotation>

</semantics>

</math><img src="./3257a6a2a3cdd7299549f0946577a7ebc3909dbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.066ex; height:3.176ex;" alt="{\textstyle |G|=p^{n}=|{\operatorname {Z} (G)}|+\sum _{i}p^{k_{i}}.}" loading="lazy"> From this we see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> must divide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\operatorname {Z} (G)}|,}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |{\operatorname {Z} (G)}|,}</annotation>

</semantics>

</math><img src="./377f8de80822d84e5b3ca703b509ef82db8af92c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.997ex; height:2.843ex;" alt="{\displaystyle |{\operatorname {Z} (G)}|,}" loading="lazy"> so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\operatorname {Z} (G)|>1.}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\operatorname {Z} (G)|>1.}</annotation>

</semantics>

</math><img src="./08aaba086ccfae6d851ec128ae6c8ebd2d40af8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.645ex; height:2.843ex;" alt="{\displaystyle |\operatorname {Z} (G)|>1.}" loading="lazy">

In particular, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4014eb4f6564eccaacc683fc9ff882da482c5268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.302ex; height:2.509ex;" alt="{\displaystyle n=2,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is an abelian group since any non-trivial group element is of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{2}.}</annotation>

</semantics>

</math><img src="./fbe6f6f7e6a6172cb4caf2bd3c41593f3f9af005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.96ex; height:3.009ex;" alt="{\displaystyle p^{2}.}" loading="lazy"> If some element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{2},}</annotation>

</semantics>

</math><img src="./45c2028709bed988125c11651a8b78b5ee0d9c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.96ex; height:3.009ex;" alt="{\displaystyle p^{2},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is isomorphic to the <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{2},}</annotation>

</semantics>

</math><img src="./45c2028709bed988125c11651a8b78b5ee0d9c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.96ex; height:3.009ex;" alt="{\displaystyle p^{2},}" loading="lazy"> hence abelian. On the other hand, if every non-trivial element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./393fcf18074cb42eafb26b76c515a1e93e17512c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p,}" loading="lazy"> hence by the conclusion above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\operatorname {Z} (G)|>1,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\operatorname {Z} (G)|>1,}</annotation>

</semantics>

</math><img src="./63be3974e38f9903a580df224b7849e2ac366b3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.645ex; height:2.843ex;" alt="{\displaystyle |\operatorname {Z} (G)|>1,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\operatorname {Z} (G)|=p>1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\operatorname {Z} (G)|=p>1}</annotation>

</semantics>

</math><img src="./7a321af4fa106c2613015d0a4c1303bea3a7f735.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.266ex; height:2.843ex;" alt="{\displaystyle |\operatorname {Z} (G)|=p>1}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{2}.}</annotation>

</semantics>

</math><img src="./fbe6f6f7e6a6172cb4caf2bd3c41593f3f9af005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.96ex; height:3.009ex;" alt="{\displaystyle p^{2}.}" loading="lazy"> We only need to consider the case when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\operatorname {Z} (G)|=p>1,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\operatorname {Z} (G)|=p>1,}</annotation>

</semantics>

</math><img src="./bc44d35adf5afcc3cb3bc8c085930c43954af6b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.913ex; height:2.843ex;" alt="{\displaystyle |\operatorname {Z} (G)|=p>1,}" loading="lazy"> then there is an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> which is not in the center of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy"> Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {C} _{G}(b)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {C} _{G}(b)}</annotation>

</semantics>

</math><img src="./bb1bf2b12e55a7a4371f0369f276696e2170391b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.009ex; height:2.843ex;" alt="{\displaystyle \operatorname {C} _{G}(b)}" loading="lazy"> includes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> and the center which does not contain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> but at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> elements. Hence the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {C} _{G}(b)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {C} _{G}(b)}</annotation>

</semantics>

</math><img src="./bb1bf2b12e55a7a4371f0369f276696e2170391b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.009ex; height:2.843ex;" alt="{\displaystyle \operatorname {C} _{G}(b)}" loading="lazy"> is strictly larger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./393fcf18074cb42eafb26b76c515a1e93e17512c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p,}" loading="lazy"> therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\operatorname {C} _{G}(b)\right|=p^{2},}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|\operatorname {C} _{G}(b)\right|=p^{2},}</annotation>

</semantics>

</math><img src="./9451097bfeb6118a846a4602a7ad3425375b91d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.272ex; height:3.176ex;" alt="{\displaystyle \left|\operatorname {C} _{G}(b)\right|=p^{2},}" loading="lazy"> therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is an element of the center of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}" loading="lazy"> a contradiction. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is abelian and in fact isomorphic to the direct product of two cyclic groups each of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./88532f4eab1d4cef71ef96c0f8c98cac36fd9257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Conjugacy_of_subgroups_and_general_subsets">Conjugacy of subgroups and general subsets</h2></div>

More generally, given any <a href="Subset" title="Subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./661adc96925066de10003e726886d8238cea7302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.424ex; height:2.343ex;" alt="{\displaystyle S\subseteq G}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> not necessarily a subgroup), define a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\subseteq G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8e1a18be292edceb145cdf85ee6afb495f6f2aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.561ex; height:2.343ex;" alt="{\displaystyle T\subseteq G}" loading="lazy"> to be conjugate to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> if there exists some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=gSg^{-1}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./615d5233bed4194bd6f22477bebdd7b93bdf257c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.448ex; height:3.009ex;" alt="{\displaystyle T=gSg^{-1}.}" loading="lazy"> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} (S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} (S)}</annotation>

</semantics>

</math><img src="./7eaaf08ba7c407d3c892ff1ed59cb745d9e71e9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.633ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} (S)}" loading="lazy"> be the set of all subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\subseteq G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8e1a18be292edceb145cdf85ee6afb495f6f2aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.561ex; height:2.343ex;" alt="{\displaystyle T\subseteq G}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is conjugate to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./23bbb1f0f6ebdfa78b4fed06049640f7386bb44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle S.}" loading="lazy">

A frequently used theorem is that, given any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d90efaede301e819c1ead4970550ad188f3781bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.071ex; height:2.509ex;" alt="{\displaystyle S\subseteq G,}" loading="lazy"> the <a href="Index_of_a_subgroup" title="Index of a subgroup">index</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {N} (S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {N} (S)}</annotation>

</semantics>

</math><img src="./7f5ec8e0f9985530be79a5696bc86b4bbc6a6b25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.052ex; height:2.843ex;" alt="{\displaystyle \operatorname {N} (S)}" loading="lazy"> (the <a href="Normalizer" class="mw-redirect" title="Normalizer">normalizer</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">) in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> equals the cardinality of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cl} (S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cl} (S)}</annotation>

</semantics>

</math><img src="./7eaaf08ba7c407d3c892ff1ed59cb745d9e71e9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.633ex; height:2.843ex;" alt="{\displaystyle \operatorname {Cl} (S)}" loading="lazy">:

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |{\operatorname {Cl} (S)}|=[G:N(S)].}</annotation>

</semantics>

</math>

This follows since, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h\in G,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5e6ce4a1096d542f74d2f9de94412ade47b1f798.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.803ex; height:2.509ex;" alt="{\displaystyle g,h\in G,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gSg^{-1}=hSh^{-1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a4b889438c962328341e23daf11f9cd254c522fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.675ex; height:3.009ex;" alt="{\displaystyle gSg^{-1}=hSh^{-1}}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{-1}h\in \operatorname {N} (S),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}h\in \operatorname {N} (S),}</annotation>

</semantics>

</math><img src="./d05051050afcd4630f706de72d3f254dc5a66a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.329ex; height:3.176ex;" alt="{\displaystyle g^{-1}h\in \operatorname {N} (S),}" loading="lazy"> in other words, if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g{\text{ and }}h}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g{\text{ and }}h}</annotation>

</semantics>

</math><img src="./f8bb2c467b8c6c78e3b37570988e7a82e87ed42c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.364ex; height:2.509ex;" alt="{\displaystyle g{\text{ and }}h}" loading="lazy"> are in the same <a href="Coset" title="Coset">coset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {N} (S).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {N} (S).}</annotation>

</semantics>

</math><img src="./0a08b354e1151f4c753093ec6514fabe8cbff2c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.698ex; height:2.843ex;" alt="{\displaystyle \operatorname {N} (S).}" loading="lazy">

By using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{a\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\{a\},}</annotation>

</semantics>

</math><img src="./6d4f88028a5a890635db3153fdca1f939b4c1f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.799ex; height:2.843ex;" alt="{\displaystyle S=\{a\},}" loading="lazy"> this formula generalizes the one given earlier for the number of elements in a conjugacy class.

The above is particularly useful when talking about subgroups of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc645a5b7e8a2022ad70fc42dbda04c008a33a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}" loading="lazy"> The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate.

Conjugate subgroups are <a href="Group_isomorphism" title="Group isomorphism">isomorphic</a>, but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.

<div class="mw-heading mw-heading2"><h2 id="Geometric_interpretation">Geometric interpretation</h2></div>

Conjugacy classes in the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of a <a href="Path-connected" class="mw-redirect" title="Path-connected">path-connected</a> <a href="Topological_space" title="Topological space">topological space</a> can be thought of as equivalence classes of <a href="Free_loop" title="Free loop">free loops</a> under free homotopy.

<div class="mw-heading mw-heading2"><h2 id="Conjugacy_class_and_irreducible_representations_in_finite_group">Conjugacy class and irreducible representations in finite group</h2></div>

In any <a href="Finite_group" title="Finite group">finite group</a>, the number of nonisomorphic <a href="Irreducible_representations" class="mw-redirect" title="Irreducible representations">irreducible representations</a> over the <a href="Complex_number" title="Complex number">complex numbers</a> is precisely the number of conjugacy classes.

<ul><li><a href="Topological_conjugacy" title="Topological conjugacy">Topological conjugacy</a> – Concept in topology</li>

<li><a href="FC-group" title="FC-group">FC-group</a> – Group in group theory mathematics</li>

<li><a href="Conjugacy-closed_subgroup" title="Conjugacy-closed subgroup">Conjugacy-closed subgroup</a></li></ul>

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

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</style><cite id="CITEREFDummitFoote2004" class="citation book cs1">Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-43334-9</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFLang2002" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (2002). Algebra. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95385-X</bdi>.</cite>

</li>

<li id="cite_note-Grillet-2007-p56-3"><a href="#cite_ref-Grillet-2007-p56_3-0">^</a> Grillet (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LJtyhu8-xYwC&pg=PA56&dq=%22the+orbits+are+the+conjugacy+classes%22">p. 56</a>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Grillet (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LJtyhu8-xYwC&pg=PA57&dq=%22The+Class+Equation%22">p. 57</a>

</li>

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

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

<ul><li><cite id="CITEREFGrillet2007" class="citation book cs1">Grillet, Pierre Antoine (2007). Abstract algebra. Graduate texts in mathematics. Vol. 242 (2 ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-71567-4</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Conjugate_elements">"Conjugate elements"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-26" href="https://en.wikipedia.org/wiki/?title=Conjugacy_class&oldid=1297472005">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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