Quotient groups and normal subgroups

Question

I am wondering if there is some characterization of the normal subgroups of a quotient group. More precisely let $G$ be a group and $H$ a normal subgroup. Let $U$ be a normal subgroup of the quotient group $G/H$. Is it possible to relate $U$ (the normal subgroups of the quotient) to the normal subgroups of $G$ or vice-versa, i.e. is there any link between them?

Thank you in advance!

Answer 1

Let $\phi$ be the quotient map from $G$ to $G/H$ then;

You can easily show that;

If $N$ is normal in $G$ then $\phi(N)$ is normal in $G/H$ and if $L$ is normal in $G/H$ then $\phi^{-1}(L)$ is normal in $G$.

Actually there is a one to one correspndonce between normal sobgroup of $G/H$ and normal subgroup of $G$ containing $H$.

http://www.proofwiki.org/wiki/Correspondence_Theorem_(Group_Theory)

Answer 2

All subgroups $U$ of $G/H$ are of the form $U = K/H$ for $H \leq K \leq G$. Furthermore, $U = K/H \unlhd G/H \iff K \unlhd G$.