3. Let $\phi: G ightarrow G^{\prime) $ be a group homomorphism and let $H$ be a subgroup of $G$. Prove that a) i) $\phi\left(g^{n} ight)=(\phi(g))^{n} .$ ii) if $1g1=n$, then $[\phi(g)[$ divides $n$. iii) $\phi(H)$ is a subgroup of $G^{\prime} $. iv) if $H$ is cyclic, then $\phi(H)$ is also cyclic. v) if $H$ is normal in $G$, then $\phi(H)$ is normal in $G^{\prime} $. b) i) A finite group $G$ has elements of order $n$ and $a$. where $p$ and CS.VS. 1390