Chapter 3.6, Problem 14E

Let $\varphi$ be a homomorphism from the group $G$ to the group $G\text{'}$.

Prove part a of Theorem $3.32$: If $e$ denotes the identity in $G$ and $e\text{'}$ denotes the identity in $G\text{'}$, then $\varphi (e)=e\text{'}$.

Prove part b of Theorem $3.32$: $\varphi (x^{-1})={[\varphi (x)]}^{-1}$ for all $x$ in $G$.