Automorphism Group/Examples

Examples of Automorphism Groups

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.

Klein $4$-Group

The automorphism group of the Klein $4$-group is the Symmetric Group on 3 Letters $S_3$.

Infinite Cyclic Group

The automorphism group of the infinite cyclic group $\Z$ is the cyclic group of order $2$.