Group theory Normal Sugbroups

Group theory Normal Sugbroups

(a) Let $N$ be a normal subgroup of a group $G$. Prove that the one-to-one
correspondence $\pi$ between the subgroups of $G$ that contain $N$ and all
of the subgroups of $G/N$ preserves normal subgroups, that is, if $K$ is a
subgroup of $G$ containing $N$, then $K$ is normal in $G$ if and only if
$\pi(K)$ is normal in $G/N$.
(b) Prove that every finite group $G$ has a homomorphic image that is a
simple group, that is, a nontrivial group with no normal subgroups other
than $\{e\}$ and itself.