---
tag: 0D6WHTTH7T
kind: definition
name: Conjugate subgroup
msc: ["20E45"]
defines: 0H11ZPX4ND
uses: [0N0PXCTYA7, 0D1VWQGZZ7]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugacy_class
    locator: Conjugacy of subgroups and general subsets
---

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The conjugate of $H$ by $g$ is the set $gHg^{-1} = \{ghg^{-1} : h \in H\}$.
Two subgroups $H$ and $K$ of $G$ are conjugate when $K = gHg^{-1}$ for some $g \in G$.

## Remarks

The set $gHg^{-1}$ is a subgroup of $G$, and $h \mapsto ghg^{-1}$ is an isomorphism from $H$ to $gHg^{-1}$.
Conjugacy of subgroups is an equivalence relation on the set of subgroups of $G$.
A subgroup that equals each of its conjugates is a normal subgroup.

## Explanations

### L1 human

Conjugating a subgroup moves it to another position inside the group, but keeps its shape.
In the symmetric group $S_3$, the three subgroups of order $2$ are conjugate to each other.
Conjugate subgroups have the same order and the same structure.