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
Canonical name	ClassEquation
Date of creation	2013-03-22 13:10:41
Last modified on	2013-03-22 13:10:41
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Theorem
Classification	msc 20E45
Synonym	conjugacy class formula
Related topic	ConjugacyClass