Abstract Algebra: Theory and Applications

Consider the function $\varphi :{\mathbb{Z}}_{10}\to {\mathbb{Z}}_{10}$ defined by $\varphi (x)=x+x\text{.}$ Prove that $\varphi$ is a group homomorphism.

For $\varphi$ defined in the previous question, explain why $\varphi$ is not a group isomorphism.

Compare and contrast isomorphisms and homomorphisms.

Paraphrase the First Isomorphism Theorem using only words. No symbols allowed at all.

“For every normal subgroup there is a homomorphism, and for every homomorphism there is a normal subgroup.” Explain the (precise) basis for this (vague) statement.