Conjugacy class

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially <a href="/facts/Group_theory/NfowcI5H">group theory</a>, two elements 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
 of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> are conjugate if there is an element 
 
 
 
 g
 
 
 {\displaystyle g}
 
 in the group such that 
 
 
 
 b
 =
 g
 a
 
 g
 
 −
 1
 
 
 .
 
 
 {\displaystyle b=gag^{-1}.}
 
 This is an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> whose <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a> are called conjugacy classes. In other words, each conjugacy class is closed under 
 
 
 
 b
 =
 g
 a
 
 g
 
 −
 1
 
 
 
 
 {\displaystyle b=gag^{-1}}
 
 for all elements 
 
 
 
 g
 
 
 {\displaystyle g}
 
 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="/facts/Non-abelian_group/KQYHrwJl">non-abelian groups</a> is fundamental for the study of their structure. For an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a>, each conjugacy class is a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> containing one element (<a href="/facts/Singleton_(mathematics)/LtNVfujT">singleton set</a>).
<a href="/facts/Function_(mathematics)/CdReEKCh">Functions</a> that are constant for members of the same conjugacy class are called <a href="/facts/Class_function/6GtYtRwN">class functions</a>.

Conjugacy class open-in-new

Conjugacy class