conjugacy class

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

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

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

$$C_x=\{y\in X:y=gx{g}^{-1}\text{for some }g\in G\}$$