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 isthe 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 ofthe distinct conjugacy classes contained in $G\setminus Z(G)$.