Groups and normal subgroups

Question

Let $H$ be normal subgroup of $G$, and $H\cap G'=\{e\}$ , $G'=\{[x,y]=xyx^{-1}y^{-1} \mid x,y\in G\}$. Prove that $\forall g \in G$ and $\forall h \in H: gh = hg$.

Answer 1

Note $gh=hg\iff ghg^{-1}h^{-1}=e$.

But, $H\triangleleft G\iff ghg^{-1}\in H$.

So, for all $g\in G,h\in H$, we have $ghg^{-1}h^{-1}\in H\cap G'=\{e\}$.