conjugacy class

Let $G$ a group, and consider its operation (action) on itself give by conjugation, that is, the mapping

(g,x)\mapsto gxg^{-1}

Since conjugation is an equivalence relation, we obtain a partition of $G$ into equivalence classes, called conjugacy classes. So, the conjugacy class of $X$ (represented $C_{x}$ or $C(x)$ is given by

C_{x}=\{y\in X:y=gxg^{-1}\mbox{for some }g\in G\}