Class Equation

A finite group decomposes into the center plus conjugacy classes of larger size

Class Equation

Class Equation. Let $G$ be a finite group . For $g \in G$ , let the conjugacy class be

\operatorname{Cl}(g)=\{xgx^{-1}: x\in G\},

and let the centralizer be

C_G(g)=\{x\in G : xg=gx\}.

Then $|\operatorname{Cl}(g)| = [G:C_G(g)]$ . If $Z(G)$ denotes the center of $G$ and $g_1,\dots,g_r$ are representatives of the distinct conjugacy classes contained in $G \setminus Z(G)$ , then

|G| = |Z(G)| + \sum_{i=1}^r [G:C_G(g_i)].

The class equation is the orbit decomposition of the conjugation action of $G$ on itself, combined with the orbit–stabilizer theorem . It is a standard tool for proving existence of normal subgroups, for example a finite p-group has nontrivial center .

Proof sketch. Under conjugation, the orbit of $g$ is $\operatorname{Cl}(g)$ and its stabilizer is $C_G(g)$ . Orbit–stabilizer gives $|\operatorname{Cl}(g)|=[G:C_G(g)]$ . Elements of $Z(G)$ have orbit size $1$ , and the remaining orbits partition $G \setminus Z(G)$ , yielding the stated sum.