Automorphism Group/Examples/Cyclic Group C3/Proof 1

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.