# 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:

 $\displaystyle \map \phi {\eqclass 0 3}$ $=$ $\displaystyle \eqclass 0 3$ $\displaystyle \map \phi {\eqclass 1 3}$ $=$ $\displaystyle \eqclass 1 3$ $\displaystyle \map \phi {\eqclass 2 3}$ $=$ $\displaystyle \eqclass 2 3$

 $\displaystyle \map \theta {\eqclass 0 3}$ $=$ $\displaystyle \eqclass 0 3$ $\displaystyle \map \theta {\eqclass 1 3}$ $=$ $\displaystyle \eqclass 2 3$ $\displaystyle \map \theta {\eqclass 2 3}$ $=$ $\displaystyle \eqclass 1 3$

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

• Results about automorphism groups can be found here.