Chapter 3.6, Problem 7E

Consider the additive groups ${\mathbb{Z}}_8$ and ${\mathbb{Z}}_4$

and define $\varphi :{\mathbb{Z}}_8\to {\mathbb{Z}}_4$

by $\varphi ({[x]}_8)={[x]}_4$.

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

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

Find ${\varphi }^{-1}(\{{[1]}_4\}),{\varphi }^{-1}(\{{[2]}_4\}),$

and ${\varphi }^{-1}(\{{[3]}_4\})$.