Automorphism Group/Examples/Cyclic Group C3/Proof 1

Example of Automorphism Group

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}$

Proof

Let $\xi$ be a general automorphism on $C_3$.

Then by Group Homomorphism Preserves Identity we immediately have that:

$\map \xi {\eqclass 0 3} = \eqclass 0 3$

Investigating $\map \xi {\eqclass 1 3}$, we find $2$ options:

$\map \xi {\eqclass 1 3} = \eqclass 1 3$

$\map \xi {\eqclass 1 3} = \eqclass 2 3$

Each leads to one and only one bijection from $C_3$ to $C_3$, that is, $\phi$ and $\theta$ as defined.

It is determined by inspection that both $\phi$ and $\theta$ are automorphisms.

Hence Automorphism Group is Subgroup of Symmetric Group is applied to confirm that $\set {\phi, \theta}$ forms a group.

The Cayley table follows.

$\blacksquare$