---
tag: 0VK5V9P8G3
kind: statement
role: proposition
name: Size of a conjugacy class is the index of the centraliser
msc: ["20E45"]
uses: [0Q88KQD17A, 0Q0Z1BEXTF, 0E2HSCKZ6C, 0P8CY9V437]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugacy_class
    locator: Conjugation as a group action, centralizers, and the class equation
---

## Statement

Let $G$ be a [[group|G5S1M9]] and let $a \in G$.
Then $|\operatorname{Cl}(a)| = [G : C_G(a)]$, where $\operatorname{Cl}(a)$ is the [[conjugacy class|0DEBXAJ5SA]] of $a$ and $C_G(a)$ is its [[centraliser|02YYV36BFM]].
In particular, if $G$ is finite, then $|\operatorname{Cl}(a)|$ divides $|G|$.

## Remarks

This is the [[orbit-stabiliser theorem|0E2HSCKZ6C]] for the [[action by conjugation|0P8CY9V437]].
Directly, $gag^{-1} = hah^{-1}$ if and only if $h^{-1}g \in C_G(a)$.
For example, in $S_3$ the centraliser of $(1\ 2\ 3)$ is $A_3$, of index $2$, and the class of $(1\ 2\ 3)$ has $2$ elements.

## Explanations

### L1 human

Conjugates of $a$ by two elements are equal exactly when the two elements differ by something that commutes with $a$.
So the conjugates of $a$ correspond to the cosets of the centraliser.
An element that commutes with many elements has few conjugates.