Conjugacy Class Equation

Theorem
Let $G$ be a group.

Let $\order G$ denote the order of $G$.

Let $\map Z G$ denote the center of $G$.

Let $x \in G$.

Let $\map {N_G} x$ denote the normalizer of $x$ in $G$.

Let $\index G {\map {N_G} x}$ denote the index of $\map {N_G} x$ in $G$.

Let $m$ be the number of non-singleton conjugacy classes of $G$.

Let $x_j: j \in \set {1, 2, \ldots, m}$ be arbitrary elements of those conjugacy classes.

Then:
 * $\displaystyle \order G = \order {\map Z G} + \sum_{j \mathop = 1}^m \index G {\map {N_G} {x_j} }$