### 1.

Consider the function $$\phi:\mathbb Z_{10}\rightarrow\mathbb Z_{10}$$ defined by $$\phi(x)=x+x\text{.}$$ Prove that $$\phi$$ is a group homomorphism.

### 2.

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

### 3.

Compare and contrast isomorphisms and homomorphisms.

### 4.

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

### 5.

“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.