---
tag: 08BPSC6YTE
kind: statement
role: theorem
name: Class equation
aliases: [class-formula]
msc: ["20E45"]
uses: [0VK5V9P8G3, 0S7Y9Q990Q, 0Q88KQD17A]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugacy_class
    locator: Conjugacy class equation
---

## Statement

Let $G$ be a finite [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.
Let $a_1, \dots, a_k$ be one element from each [[conjugacy class|0DEBXAJ5SA]] of $G$ that has more than one element.
Then $|G| = |Z(G)| + \sum_{i=1}^{k} [G : C_G(a_i)]$.

## Remarks

The conjugacy classes partition $G$, and the classes with one element are exactly the elements of $Z(G)$.
Each term $[G : C_G(a_i)]$ is the [[size of a class|0VK5V9P8G3]], so it divides $|G|$ and is greater than $1$.
For $S_3$, the equation is $6 = 1 + 3 + 2$.
The class equation is the key step in the proof that a nontrivial finite $p$-group has a nontrivial centre.

## Explanations

### L1 human

Count the elements of the group class by class.
The central elements form classes of size one, and the other classes have sizes that divide the order of the group.
This simple count gives strong facts, for example about groups of prime power order.