Knowlpedia

Class Equation

The order of a finite group is the order of its center plus the sizes of its noncentral conjugacy classes.

Let GG be a finite . For g∈Gg\in G, set

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

and let the be

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

Then ∣Cl⁡(g)∣=[G:CG(g)]|\operatorname{Cl}(g)|=[G:C_G(g)]. If g1,…,grg_1,\dots,g_r represent the noncentral , the class equation is

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

The class equation is the orbit decomposition of the of GG on itself, combined with the . It is a standard tool for proving existence of normal subgroups, for example .