Image of a normal subgroup and induced morphisms

#group_theory

Let $σ : G \to H$ be a group epimorphism. Let $N$ be a normal subgroup of $G$ and $K = σ (N)$ , the image of $N$ in $H$ .

Prove that $K$ is a normal subgroup of $H$ . Give an example to show that this is not true if $σ$ is not onto.
Under what conditions does $σ$ induce a homomorphism $G / N \to H / K$ , and when is this an isomorphism? Prove your answer.

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Let $\sigma:G\to H$ be a group epimorphism. Let $N$ be a normal subgroup of $G$ and $K=\sigma(N)$, the image of $N$ in $H$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $K$ is a normal subgroup of $H$. Give an example to show that this is not true if $\sigma$ is not onto.
	\item Under what conditions does $\sigma$ induce a homomorphism $G/N\to H/K$, and when is this an isomorphism? Prove your answer.
\end{enumerate}