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 $C_8$ can be denoted by $\eqclass 0 8, \eqclass 1 8, \ldots, \eqclass 7 8$.

By Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order, the generators of $C_8$ are exactly $\eqclass 1 8, \eqclass 3 8, \eqclass 5 8, \eqclass 7 8$.

By Automorphism Maps Generator to Generator and Homomorphism of Generated Group, there are exactly $4$ automorphisms for the group $C_8$, namely:

$\phi_1: \eqclass 1 8 \mapsto \eqclass 1 8$

$\phi_3: \eqclass 1 8 \mapsto \eqclass 3 8$

$\phi_5: \eqclass 1 8 \mapsto \eqclass 5 8$

$\phi_7: \eqclass 1 8 \mapsto \eqclass 7 8$

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By direct computation, the Cayley table for these $4$ elements can be presented as:

$\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 $4$-group.

This shows that the automorphism group of $C_8$ is isomorphic to the Klein $4$-group.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $27 \ \text {(ii)}$