The Class Equation

We note that by the Orbit Stabilizer Theorem then $| O_{x} | = | G : C_{G} (x) |$ (see Conjugacy Classes and Index (Groups) for more info). An element $x$ is in the Center of $G$ exactly when $O_{x} = {x}$ when $x$ is a fixed point:

\begin{aligned} | O_{x} | = 1 & ⟺ O_{x} = {x} \\ ⟺ g x g^{- 1} = x \forall g \in G \\ ⟺ x g = g x \forall g \in G \\ ⟺ x \in Z (G) \end{aligned}

Since $G$ is a disjoint union of its distinct conjugacy classes (ie: its orbits), the Order (Groups) of $G$ equals the sum of the sizes of the distinct conjugacy classes.

The Class Equation

Let $g_{1}, . ., g_{r} \in G$ be representatives of the distinct conjugacy classes of $G$ not contained in the center of $G$ . Then:

| G | = | Z (G) | + \sum_{i = 1}^{r} | G : C_{G} (g_{i}) |