# Definition:Automorphism Group

## Definition

### Group of Automorphisms

Let $\struct {S, *}$ be an algebraic structure.

Let $\mathbb S$ be the set of automorphisms of $S$.

Then the algebraic structure $\struct {\mathbb S, \circ}$, where $\circ$ denotes composition of mappings, is called the **automorphism group** of $S$.

The structure $\struct {S, *}$ is usually a group. However, this is not necessary for this definition to be valid.

The **automorphism group of $S$** is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\Aut S$.

## Also see

- Definition:Homeomorphism Group, which plays the role as the group of automorphisms of a topological space
- Definition:Galois Group of Field Extension
- Definition:Galois Group of Covering Map