19 - The Class Equation

conjugacy classes

In general, when $G$ acts on its own elements by conjugation, then the Orbits are called conjugacy classes. The Stabilizers under this action are exactly the Centralizers of these elements:
$G_{x} = {g \in G | g x g^{- 1} = x} = {g \in G | g x = x g} = C_{G} (x)$

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

In general, suppose $G$ acts on its collection of subgroups by conjugation and $H \leq G$ is a specific Subgroup. Then:

The subgroup $H$ is fixed under this action exactly when $H ⊴ G$
The stabilizer $G_{H}$ is the Normalizer $N_{G} (H)$ of $H$ in $G$
The number of subgroups conjugate to $H$ equals $| G : N_{G} (H) |$ .