Automorphism Group/Examples/Cyclic Group C3

Example of Automorphism Group
Consider the cyclic group $C_3$, which can be presented as its Cayley table:

The automorphism group of $C_3$ is given by:


 * $\Aut {C_3} = \set {\phi, \theta}$

where $\phi$ and $\theta$ are defined as:

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

The Cayley table follows.