Automorphism Group/Examples/Infinite Cyclic Group
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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 $27 \ \text {(iii)}$