Chapter 3.6, Problem 4E

Consider the additive group $\mathbb{Z}$

and the multiplicative group $G=\left\{1,i,-1,-i\right\}$

and define $\varphi :\mathbb{Z}\to G$

by $\varphi (n)=i^n$.

Prove that $\varphi$ is a homomorphism and find ker $\varphi$.

Is $\varphi$ an epimorphism? Is $\varphi$ a monomorphism?

Find ${\varphi }^{-1}(\{i\}),{\varphi }^{-1}(\{-1\}),{\varphi }^{-1}(\{-i\}),{\varphi }^{-1}(\{1,-1\}),$

and ${\varphi }^{-1}(\{i,-i\})$.