# Normal subgroups of a group

I have two normal subgroups (call them $X$ and $Y$) of a group with only the identity element in common and need to show that $xy = yx$ for all $x \in X, y \in Y$.
I really have no idea where to begin although I do understand the definition of a normal subgroup. I think that $gxyg^-1$ is in $XY$ but am not sure what this means, whether or not $XY$ is a normal subgroup or whether I am meant to look for a homomorphism or isomorphism.
Notice that the element $x(yx^{-1}y^{-1}) =(xyx^{-1})y^{-1}$ is in both X and Y (why?) and must therefore be the identity element.