Definition:Automorphism Group/Group

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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.


Symbol

The automorphism group of an algebraic structure $S$ 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

  • Results about automorphism groups can be found here.


Sources


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