Class equation decomposition - Imported Knowlpedia Content

Proposition (Decomposition underlying the class equation). Let $G$ be a finite group. Consider the action of $G$ on itself by conjugation (see conjugation action). Then:

$G$ is a disjoint union of its conjugacy classes.
The elements with singleton conjugacy class are exactly the center $Z(G)$ .
For each $x\in G$ , the size of the conjugacy class of $x$ equals the index $[G:C_G(x)]$ , where $C_G(x)$ is the centralizer, and this is a consequence of the orbit–stabilizer theorem.

In particular, choosing one representative $x_i$ from each conjugacy class outside the center yields the decomposition

|G| \;=\; |Z(G)| \;+\; \sum_i [G:C_G(x_i)].

Remarks

Context. This is the structural content behind the class equation: it turns the conjugation action into a counting identity, a key tool for $p$ -groups and Sylow theory.