class equation

Take a finite group and have it act on itself by conjugation. The fixed points of this action are exactly the elements that commute with everything — the center — while the rest fall into nontrivial conjugacy classes. The class equation is the bookkeeping that results: it adds up the center and these classes to recover the whole group's order.

Precisely, for a finite group G, choosing one representative g_i from each conjugacy class that lies outside the center Z(G), |G| = |Z(G)| + sum over i of [G : C_G(g_i)]. Each summand [G : C_G(g_i)] is the size of a nontrivial conjugacy class and, being an index, divides |G| and is greater than 1.

This deceptively simple identity is a workhorse. Because every nontrivial-class term is divisible by p when |G| is a power of p, the class equation forces the center of a nontrivial p-group to be nontrivial — the cornerstone of p-group theory. More generally it is the standard tool for proving the existence of central elements, deriving Sylow's theorems, and constraining the possible orders of subgroups and classes.