---
tag: 0Q88KQD17A
kind: definition
name: Conjugacy class
msc: ["20E45"]
defines: 0DEBXAJ5SA
uses: [0D1VWQGZZ7, 0ZC7FYE0NZ]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugacy_class
    locator: Definition
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $a \in G$.
The conjugacy class of $a$ is the set $\operatorname{Cl}(a) = \{gag^{-1} : g \in G\}$.

## Remarks

The conjugacy classes are the equivalence classes of the relation "is [[conjugate to|0DXD4QSZ8A]]", so they partition $G$.
They are the [[orbits|0YQPZB5Q1G]] of the [[action by conjugation|0P8CY9V437]].
The class of $a$ is $\{a\}$ if and only if $a$ lies in the [[centre|0DZ93493VN]] of $G$.
In the symmetric group $S_n$, the conjugacy classes are given by the [[cycle types|0J438GAC8K]].

## Explanations

### L1 human

The conjugacy class of an element collects all the elements that look like it after a renaming by conjugation.
In an abelian group, each element is alone in its class.
In $S_3$, the three transpositions form one class, and the two $3$-cycles form another class.