Conjugate element - Imported Knowlpedia Content

Let $G$ be a group and let $g,h\in G$ . We say that $h$ is conjugate to $g$ (in $G$ ), and write $h\sim g$ , if there exists an element $x\in G$ such that

h = x g x^{-1}.

For a fixed $x\in G$ , the map $c_x\colon G\to G$ given by $c_x(g)=xgx^{-1}$ is called conjugation by $x$ ; it is an inner automorphism of $G$ .

Remarks

Conjugacy is an equivalence relation on $G$ ; its equivalence classes are the conjugacy classes. Equivalently, conjugacy classes are the orbits of the conjugation action of $G$ on itself. Conjugation also controls normality: a normal subgroup is precisely a subgroup stable under conjugation.

Examples

If $G$ is abelian, then $xgx^{-1}=g$ for all $x,g\in G$ , so every element is conjugate only to itself.
In $S_3$ , the transpositions are all conjugate: for instance $(123)(12)(123)^{-1}=(23)$ .
In a matrix group $G\le GL_n(\mathbb{F})$ , conjugation $XAX^{-1}$ corresponds to similarity of matrices; conjugate matrices represent the same linear operator in different bases.