# If (G/N)/(H/N) abelian. Then H normal in G?

Let $G$ be a group and $N\leq H\leq G$ be two subgroups of $G$ and $N\unlhd G$. Assume $H/N\unlhd G/N$. What condition(s) on $\frac{G/N}{H/N}$ to guarantee that $H\unlhd G$?

Edit: If $\frac{G/N}{H/N}$ is abelian. Does that imply $H$ is a normal subgroup of $G$?

• $N$ must be normal in $G$, not only in $H$. – Bernard Dec 4 '15 at 21:24
• oh yes, I will edit it – Ronald Dec 4 '15 at 21:25

The Lattice Isomorphism Theorem tells us there is a correspondence between normal subgroups of the quotient group and the original group. That is, $H/N\unlhd G/N$ if and only if $H\unlhd G$, assuming the quotient is well-defined. So you should not need any other conditions.