class equation

The conjugacy classes of a group form a partition of its elements. In a finite group, this means that the order of the group is the sum of the number of elements of the distinct conjugacy classes. For an element $g$ of group $G$ , we denote the centralizer in $G$ of $g$ by $C_G(g)$ . The number of elements in the conjugacy class of $g$ is $[G:C_G(g)]$ , the index of $C_G(g)$ in $G$ . For an element $g$ of the center $Z(G)$ of $G$ , the conjugacy class of $g$ consists of the singleton $\{g\}$ . Putting this together gives us the class equation$$ |G| = |Z(G)| + \sum_{i=1}^m [G:C_G(x_i)]$$ where the $x_i$ are elements of the distinct conjugacy classes contained in $G\setminus Z(G)$ .