Editing Conjugacy class (section)

==Definition<!--'Class number (group theory)' redirects here-->==

Let <math>G</math> be a group. Two elements <math>a, b \in G</math> are '''conjugate''' if there exists an element <math>g \in G</math> such that <math>gag^{-1} = b,</math> in which case <math>b</math> is called {{em|a conjugate}} of <math>a</math> and <math>a</math> is called a conjugate of <math>b.</math>

In the case of the [[general linear group]] <math>\operatorname{GL}(n)</math> of [[invertible matrices]], the conjugacy relation is called  [[matrix similarity]].

It can be easily shown that conjugacy is an equivalence relation and therefore partitions <math>G</math> into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes <math>\operatorname{Cl}(a)</math> and <math>\operatorname{Cl}(b)</math> are equal [[if and only if]] <math>a</math> and <math>b</math> are conjugate, and [[Disjoint sets|disjoint]] otherwise.) The equivalence class that contains the element <math>a \in G</math> is
<math display="block">\operatorname{Cl}(a) = \left\{ gag^{-1} : g \in G \right\}</math>
and is called the '''conjugacy class''' of <math>a.</math> The '''{{visible anchor|class number|Class number (group theory)}}'''<!--boldface per WP:R#PLA--> of <math>G</math> is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same [[Order (group theory)|order]].

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 [[symmetric group]] they can be described by [[Permutation#Cycle type|cycle type]].