I have two normal subgroups (call them $\displaystyle X$ and $\displaystyle Y$) of a group with only the identity element in common and need to show that $\displaystyle xy = yx$ for all $\displaystyle x \in X, y \in Y$.

I really have no idea where to begin although I do understand the definition of a normal subgroup. I think that $\displaystyle gxyg^-1$ is in $\displaystyle XY$ but am not sure what this means, whether or not $\displaystyle XY$ is a normal subgroup or whether I am meant to look for a homomorphism or isomorphism.

Any help appreciated, thanks.