Conjugate element

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 $G$.

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.