# 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 $G\setminus Z(G)$.

Title class equation ClassEquation 2013-03-22 13:10:41 2013-03-22 13:10:41 yark (2760) yark (2760) 9 yark (2760) Theorem msc 20E45 conjugacy class formula ConjugacyClass