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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 $G$?

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

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