Automorphism Group/Examples/Cyclic Group C3

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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 1

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.

$\blacksquare$


Proof 2

This is an example of Order of Automorphism Group of Prime Group:

$\order {\Aut G} = p - 1$

for a group of prime order $p$.

The only group of order $2$ is the cyclic group of order $2$.

The result follows.

$\blacksquare$


Sources

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