I'm quite new to this whole topic and so I don't know how get a grip on this question:
Let $G$ be a group and $U,V$ two subgroups. Denote by $[U,V]$ the subgroup of $G$ generated by $[u,v]=uvu^{-1}v^{-1}$ with $u \in U, v \in V$. Show: $U\times V \rightarrow G$, $(u,v) \mapsto [u,v]$ is bilinear $\Leftrightarrow$ [U,V] is contained in the centre of the subgroup of $G$, that is generated by $U$ and $V$
I'd be happy for any hint and help :)