Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G? Suppose $G$ is a non-abelian group and $H,K$ are two abelian subgroups of $G$. Then must $HK$ be an abelian subgroup of $G$?
I know an example, but I am confused. Thus I just want to check that.

This is not true. There are even metabelian groups (non-abelian, but $G'$ abelian) which are the product of two abelian subgroups $A$ and $B$, i.e., $G=AB$.
