Conjugate elements

This article describes an equivalence relation on the set of elements of a group

Definition

Symbol-free definition

Given a group, two (possibly equal) elements of the group are termed conjugate elements if there is an inner automorphism of the group mapping one element to the other.

Definition with symbols

Given a group $G$ and elements $g,h\in G$, $g$ is termed conjugate to $h$ if there exists $x\in G$ such that $xg{x}^{-1}=h$, in other words, the inner automorphism of conjugation by $x$, sends $g$ to $h$.

The equivalence classes under the equivalence relation of being conjugate are termed the conjugacy classes.