Conjugacy Class Equation/Proof 2

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 $G$.

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

Proof
Let the distinct orbits of $G$ under the Conjugacy Action be $\operatorname{Orb} \left({x_1}\right), \operatorname{Orb} \left({x_2}\right), \ldots, \operatorname{Orb} \left({x_s}\right)$.

Then from the Partition Equation:
 * $\left|{G}\right| = \left|{\operatorname{Orb} \left({x_1}\right)}\right| + \left|{\operatorname{Orb} \left({x_2}\right)}\right| + \cdots + \left|{\operatorname{Orb} \left({x_s}\right)}\right|$

From the Orbit-Stabilizer Theorem:
 * $\left|{\operatorname{Orb} \left({x_i}\right)}\right| \backslash \left|{G}\right|, i = 1, \ldots, s$

The result follows from the definition of the Conjugacy Action.