# Definition:Automorphism Group/Group

## 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:

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

## Examples

### Cyclic Group $C_3$

Consider the cyclic group $C_3$, which can be presented as its Cayley table:

- $\begin{array}{r|rrr} \struct {\Z_3, +_3} & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 1 3 & \eqclass 1 3 & \eqclass 2 3 & \eqclass 0 0 \\ \eqclass 2 3 & \eqclass 2 3 & \eqclass 0 3 & \eqclass 1 3 \\ \end{array}$

The automorphism group of $C_3$ is given by:

- $\Aut {C_3} = \set {\phi, \theta}$

where $\phi$ and $\theta$ are defined as:

\(\ds \map \phi {\eqclass 0 3}\) | \(=\) | \(\ds \eqclass 0 3\) | ||||||||||||

\(\ds \map \phi {\eqclass 1 3}\) | \(=\) | \(\ds \eqclass 1 3\) | ||||||||||||

\(\ds \map \phi {\eqclass 2 3}\) | \(=\) | \(\ds \eqclass 2 3\) |

\(\ds \map \theta {\eqclass 0 3}\) | \(=\) | \(\ds \eqclass 0 3\) | ||||||||||||

\(\ds \map \theta {\eqclass 1 3}\) | \(=\) | \(\ds \eqclass 2 3\) | ||||||||||||

\(\ds \map \theta {\eqclass 2 3}\) | \(=\) | \(\ds \eqclass 1 3\) |

The Cayley table of $\Aut {C_3}$ is then:

- $\begin{array}{r|rr} & \phi & \theta \\ \hline \phi & \phi & \theta \\ \theta & \theta & \phi \\ \end{array}$

### Cyclic Group $C_8$

The automorphism group of the cyclic group $C_8$ is the Klein $4$-group.

## Also see

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

- Results about
**automorphism groups**can be found**here**.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64 \alpha$ - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\S 1.2$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $24$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Proposition $8.11$