---
tag: DSQ17TNW1C
kind: concept
name: Automorphism
msc: [08A35, 03C07]
specializes: [PZMGQMDN25]
sources:
  - url: https://en.wikipedia.org/wiki/Automorphism
same_as:
  wikidata: Q782566
  wikipedia: Automorphism
---

## Summary

An automorphism of a [[structure|983GZBQYT4]] is an [[isomorphism|PZMGQMDN25]] from the structure to itself.
It is a symmetry of the structure: a permutation of its elements that keeps all operations and relations.
The automorphisms of a structure form a group under composition, the [[automorphism group|0K1SCPBNKJ]].
Complex conjugation, as an automorphism of the field $\C$, and the [[Frobenius automorphism of a finite field|0CGCPX8Z70]] are classical examples.

## Explanations

### L1 human

An automorphism is a way to shuffle the elements of a structure so that nobody can tell the difference afterwards.
For the integers with addition, the map $x \mapsto -x$ is an automorphism: it turns sums into sums and $0$ into $0$.
For a square drawn as a graph with four corners and four sides, each rotation and reflection of the square gives an automorphism.
The identity map is always an automorphism.