Automorphism Group/Examples/Cyclic Group C8

Example of Automorphism Group
The automorphism group of the cyclic group $C_8$ is the Klein $4$-group.

Proof
The cyclic group $C_8$ is isomorphic to the underlying group of the ring $\Z_8$.

So the elements of $C_8$ could be denoted by $\eqclass{0}{8}, \eqclass{1}{8}, \ldots, \eqclass{7}{8}$.

By Generators of Finite Cyclic Group, the generators are exactly $\eqclass{1}{8}, \eqclass{3}{8}, \eqclass{5}{8}, \eqclass{7}{8}$.

By Automorphism maps Generators to Generators and Group Homorphism determined by Image of Generators, there are exactly 4 automorphism for the group $C_8$, namely:


 * $\phi_1: \eqclass{1}{8} \mapsto \eqclass{1}{8}$
 * $\phi_3: \eqclass{1}{8} \mapsto \eqclass{3}{8}$
 * $\phi_5: \eqclass{1}{8} \mapsto \eqclass{5}{8}$
 * $\phi_7: \eqclass{1}{8} \mapsto \eqclass{7}{8}$

By direct computation, the multiplication table for these 4 elements could be drawn:


 * $\begin{array}{c|cccc}

& \phi_1 & \phi_3 & \phi_5 & \phi_7 \\ \hline \phi_1 & \phi_1 & \phi_3 & \phi_5 & \phi_7 \\ \phi_3 & \phi_3 & \phi_1 & \phi_7 & \phi_5 \\ \phi_5 & \phi_5 & \phi_7 & \phi_1 & \phi_3 \\ \phi_7 & \phi_7 & \phi_5 & \phi_3 & \phi_1 \\ \end{array}$

which is the same as the Cayley table of the Klein-Four group.

This shows that the automorphism group of $C_8$ is isomorphic to the Klein-Four group.