Homomorphism to abelian group

Question

Let $\phi :G \rightarrow G'$ be group homomorphism and $H$ is subgroup of $G$ such that $ker(\phi)\subseteq H$. Also $G'$ is abelian. Then which of the following are correct.

• H is normal in G

• $ker(\phi)=H$

• H is not normal in G

I know all subgroups of G' is normal in G' and pre image of normal subgroup is normal in G. Will this imply H normal in G

Answer 1

Yes. $\phi(H)$ is a subgroup of $G'$. All subgroups of an abelian group are normal. Therefore the preimage of $\phi(H)$ is normal in $G$.

Answer 2

The fact that $H$ is normal in $G$ is a direct consequence of the correspondence theorem of groups.

Actually, since $H$ contains $\ker\phi$ and $f(H)$ is normal in $G’$(Since G’ is abelian), apply the theorem we have $H$ is normal in $G$.