I thought this one would be easier than it has turned out to be. So given a group $G$ there exists some group $N\triangleleft G$ such that both $N$ and $G/N$ are abelian. I considered some arbitrary subgroup $K\subseteq G$ and then my instinct was to look at the group $K\cap N$ which is abelian in $G$ and both abelian and normal in $K$, but then I hit a snag looking at $K\;/\;K\cap N$.
If $K\subseteq N$ or $N\subseteq K$, then the proof is easy, but what happens when their intersection is not just $K$ or $N$, but smaller? possibly just the identity even. Neither the correspondence between subgroups of $G$ containing $N$ and subgroups of $G/N$, nor the 1st or 3rd isomorphism theorems seem to be of much help (the 2nd iso theorem is given as a later exercise and thus shouldn't be needed).
Can anyone help me with this? Thanks.