---
tag: 01CS8Q2APC
kind: definition
name: Characteristic subgroup
msc: [20F28]
defines: 0KABPMRHM2
uses: [0CWXCWP0Y0, 0N0PXCTYA7]
sources:
  - url: https://en.wikipedia.org/wiki/Characteristic_subgroup
    locator: Definition
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The subgroup $H$ is characteristic in $G$ when $\alpha(H) = H$ for each [[automorphism|0K1SCPBNKJ]] $\alpha$ of $G$.

## Remarks

It is equivalent to ask $\alpha(H) \subseteq H$ for each automorphism $\alpha$, because then also $\alpha^{-1}(H) \subseteq H$.
The [[centre|0DZ93493VN]] and the [[commutator subgroup|0ASGSTXSBW]] of a group are characteristic.
A characteristic subgroup of a normal subgroup of $G$ is normal in $G$.

## Explanations

### L1 human

A characteristic subgroup is a subgroup that no symmetry of the group can move.
Usually such a subgroup has a description that uses only the group operation, like "the elements that commute with everything".
Any renaming of the elements that keeps the products correct must keep such a subgroup in place.