Editing Conjugacy class (section)

==Properties==

* The [[identity element]] is always the only element in its class, that is <math>\operatorname{Cl}(e) = \{ e \}.</math>
* If <math>G</math> is [[abelian group|abelian]] then <math>gag^{-1} = a</math> for all <math>a, g \in G</math>, i.e. <math>\operatorname{Cl}(a) = \{ a \}</math> for all <math>a \in G</math> (and the converse is also true: if all conjugacy classes are singletons then <math>G</math> is abelian).
* If two elements <math>a, b \in G</math> belong to the same conjugacy class (that is, if they are conjugate), then they have the same [[Order (group theory)|order]]. More generally, every statement about <math>a</math> can be translated into a statement about <math>b = gag^{-1},</math> because the map <math>\varphi(x) = gxg^{-1}</math> is an [[group isomorphism#Automorphisms|automorphism]] of <math>G</math> called an [[inner automorphism]]. See the next property for an example.
* If <math>a</math> and <math>b</math> are conjugate, then so are their powers <math>a^k</math> and <math>b^k.</math> (Proof: if <math>a = gbg^{-1}</math> then <math>a^k = \left(gbg^{-1}\right)\left(gbg^{-1}\right) \cdots \left(gbg^{-1}\right) = gb^kg^{-1}.</math>) Thus taking {{mvar|k}}th 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>a</math> is a power-up class of <math>a^k</math>).
* An element <math>a \in G</math> lies in the [[Center of a group|center]] <math>\operatorname{Z}(G)</math> of <math>G</math> if and only if its conjugacy class has only one element, <math>a</math> itself. More generally, if <math>\operatorname{C}_G(a)</math> denotes the {{em|[[centralizer]]}} of <math>a \in G,</math> i.e., the [[subgroup]] consisting of all elements <math>g</math> such that <math>ga = ag,</math> then the [[Index of a subgroup|index]] <math>\left[G : \operatorname{C}_G(a)\right]</math> is equal to the number of elements in the conjugacy class of <math>a</math> (by the [[orbit-stabilizer theorem]]).
* Take <math>\sigma \in S_n</math> and let <math>m_1, m_2, \ldots, m_s</math> be the distinct integers which appear as lengths of cycles in the cycle type of <math>\sigma</math> (including 1-cycles). Let <math>k_i</math> be the number of cycles of length <math>m_i</math> in <math>\sigma</math> for each <math>i = 1, 2, \ldots, s</math> (so that <math>\sum\limits_{i=1}^s k_i m_i = n</math>). Then the number of conjugates of <math>\sigma</math> is:<ref name="dummit" /><math display="block">\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)}.</math>