Conjugacy class (reviewed)

If $G$ is a group and $x\in G$, the conjugacy class of $x$ is the set $\{gxg^{-1}\mid g\in G\}$. These sets partition $G$, and the set of conjugacy classes is denoted by $\mathrm{conj}(G)$.

Since conjugation by fixed $g\in G$ is an automorphism of $G$, all conjugate elements have the same order in the group.

If $G\leq S_n$, then all elements in the conjugacy class of an element have the same cycle type.