Conjugacy Class Equation

Theorem
Let $G$ be a group.

Let $\left|{G}\right|$ be the order of $G$.

Let $Z \left({G}\right)$ be the center of $G$.

Let $x \in G$.

Let $N_G \left({x}\right)$ be the normalizer of $x$ in $G$.

Let $\left[{G : N_G \left({x}\right)}\right]$ be the index of $N_G \left({x}\right)$ in $G$.

Let $m$ be the number of non-singleton conjugacy classes of $G$, and let ${x_j}, j \in \{1, \ldots, m\}$ be representatives of those conjugacy classes.

Then:
 * $\displaystyle \left|{G}\right| = \left|{Z \left({G}\right)}\right| + \sum_{j \mathop = 1}^m \left[{G : N_G \left({x_j}\right)}\right]$