Chapter 3.6, Problem 6E

Consider the additive groups ${\mathbb{Z}}_{12}$

and ${\mathbb{Z}}_6$

and define $\varphi :{\mathbb{Z}}_{12}\to {\mathbb{Z}}_6$

by $\varphi ({[x]}_{12})={[x]}_6$.

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

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

Find ${\varphi }^{-1}(\{{[1]}_6\}),{\varphi }^{-1}(\{{[2]}_6\}),$

and ${\varphi }^{-1}(\{{[4]}_6\})$.