Editing Conjugacy class (section)

{{short description|In group theory, equivalence class under the relation of conjugation}}
[[File:Dihedral-conjugacy-classes.svg|thumb|420px|Two [[Cayley graph]]s of [[dihedral group]]s with conjugacy classes distinguished by color.]]

In [[mathematics]], especially [[group theory]], two elements <math>a</math> and <math>b</math> of a [[Group (mathematics)|group]] are '''conjugate''' if there is an element <math>g</math> in the group such that <math>b = gag^{-1}.</math> This is an [[equivalence relation]] whose [[equivalence class]]es are called '''conjugacy classes'''.  In other words, each conjugacy class is closed under <math>b = gag^{-1}</math> for all elements <math>g</math> 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 [[non-abelian group]]s is fundamental for the study of their structure.<ref name="dummit">{{cite book|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|title=Abstract Algebra|publisher=[[John Wiley & Sons]]|year=2004|edition=3rd|isbn=0-471-43334-9}}</ref><ref>{{cite book|last=Lang|first=Serge|author-link=Serge Lang|title=Algebra|publisher=[[Springer Science+Business Media|Springer]]|series=[[Graduate Texts in Mathematics]]|year=2002|isbn=0-387-95385-X}}</ref> For an [[abelian group]], each conjugacy class is a [[Set (mathematics)|set]] containing one element ([[Singleton (mathematics)|singleton set]]).

[[function (mathematics)|Function]]s that are constant for members of the same conjugacy class are called [[class function]]s.