Let $G$ be a group. For any $a$, $g \in G$, the conjugation of $a$ by $g$ is
\[\conj_g a = gag^{-1}.\]For any $a$, $b \in G$, $a$ and $b$ are conjugate, written
\[a \stackrel{\conj}{\sim} b,\]if there exists $g\in G$ such that
\[\conj_g a = b.\]
For any group, $\stackrel{\conj}{\sim}$ is an equivalence relation.