Class equation of a group

Statement

Suppose $G$ is a finite group, $Z(G)$ is the center of $G$, and $c_{1},c_{2},\dots ,c_{r}$ are all the conjugacy classes in $G$ comprising the elements outside the center. Let $g_{i}$ be an element in $c_{i}$ for each $1\leq i\leq r$. Then, we have:

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

Note that this is a special case of the class equation of a group action where the group acts on itself by conjugation.

Related facts

• Class equation of a group relative to a prime power

Facts used

1. Class equation of a group action

Proof

The proof follows directly from fact (1), and the following observations:

• When a group acts on itself by conjugation, the set of fixed points under the action is precisely the center of the group.
• The stabilizer of a point $g_{i}$ under the action by conjugation is precisely the centralizer of $g_{i}$.