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 $G$?

  • $\begingroup$ $N$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.