---
tag: 0SRE7K4E49
kind: statement
role: proposition
name: Characteristic subgroups are normal
msc: [20F28]
uses: [01CS8Q2APC, 0H47DMFN3S, 0W77CVBCHQ]
sources:
  - url: https://en.wikipedia.org/wiki/Characteristic_subgroup
    locator: Normal subgroup
---

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[characteristic subgroup|0KABPMRHM2]] of $G$.
Then $H$ is a [[normal subgroup|0CB0DX5BVV]] of $G$.

## Remarks

Each [[inner automorphism|0RN00NTM3S]] $c_g$ is an automorphism, so $gHg^{-1} = c_g(H) = H$ for each $g \in G$.
The converse is false: in the Klein four-group $V = \{e, a, b, ab\}$, each subgroup is normal, but an automorphism can swap $\{e, a\}$ and $\{e, b\}$.

## Explanations

### L1 human

Normal means fixed by the inner automorphisms, and characteristic means fixed by all automorphisms.
All automorphisms include the inner ones.
So a characteristic subgroup is normal.