---
tag: 0CWXCWP0Y0
kind: definition
name: Automorphism group
aliases: [group-of-automorphisms]
msc: [20F28]
defines: 0K1SCPBNKJ
uses: [085PND4KNT]
sources:
  - url: https://en.wikipedia.org/wiki/Automorphism_group
    locator: lead section
---

## Definition

Let $G$ be a [[group|G5S1M9]].
An automorphism of $G$ is a [[group isomorphism|08MG7H8JW5]] from $G$ to $G$.
The automorphism group $\operatorname{Aut}(G)$ is the set of all automorphisms of $G$, with composition of maps as the operation.

## Remarks

The identity map is the identity element of $\operatorname{Aut}(G)$, and the inverse of an automorphism is its inverse map.
The automorphism group of $(\mathbb{Z}, +)$ has two elements, the identity and $n \mapsto -n$.
For an [[abelian group|082GHNPJ53]] $G$, the map $g \mapsto g^{-1}$ is an automorphism.

## Explanations

### L1 human

An automorphism is a way to rename the elements of a group so that all products stay correct.
The set of all such renamings is again a group, because you can compose and undo renamings.
For the integers, the only renamings are "do nothing" and "change all signs".