Automorphism Group/Examples/Infinite Cyclic Group

Example of Automorphism Group

The automorphism group of the infinite cyclic group $\Z$ is the cyclic group of order $2$.

Proof

Let $g$ be a generator of $\Z$.

Let $\varphi$ be an automorphism on $\Z$.

By Homomorphic Image of Cyclic Group is Cyclic Group, $\map \varphi g$ is a generator of $\Z$.

By Homomorphism of Generated Group, $\varphi$ is uniquely determined by $\map \varphi g$.

By Generators of Infinite Cyclic Group, there are $2$ possible values for $\map \varphi g$.

Therefore there are $2$ automorphisms on $\Z$:

$\order {\Aut \Z} = 2$

By Prime Group is Cyclic, the automorphism group of the infinite cyclic group $\Z$ is the cyclic group of order $2$.

$\blacksquare$

Sources

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