Conjugation

Conjugation by $g$ is the group automorphism $x\mapsto gx{g}^{-1}$. Its orbits are the conjugacy classes.

Two group elements $x$ and $y$ are conjugate when $y=gx{g}^{-1}$ for some group element $g$, equivalently when they belong to the same conjugacy class. Conjugate elements have the same order of a group element.