# Definition:Automorphism Group/Group

< Definition:Automorphism Group(Redirected from Definition:Group of Automorphisms)

## Contents

## Definition

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 known as

The **automorphism group** is also known as the **group of automorphisms**.

The **automorphism group** of $S$ can be found denoted in a number of ways, for example:

- $\map {\mathscr A} S$
- $\map A S$

## Also see

- Group of Automorphisms is Subgroup of Symmetric Group, where it is also demonstrated that $\Aut S$ is actually a group.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 64 \alpha$ - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\S 1.2$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\S 8$: Proposition $8.11$