Coset Proof. Looking for ideas

The overall problem is this:

Quote:

If H < N and H, N, and K are subgroups of G, then $\displaystyle HK \cap N = H( K \cap N)$.

I have tried various and sundry methods to work with this, to little result. I have managed to prove that $\displaystyle HK \cap N$ is a subgroup of G, but that was from a tactic that I eventually had to consider unfruitful.

Part of my puzzlement is that the only sign of the existence of G in any of the work I have done is that H, N, and K all have the same identity element. Notable perhaps, but I had expected G to show up more prominently.

I also had the thought that a possible generalization of this theorem exists: $\displaystyle HK \cap HN = H(K \cap N)$. (H is merely a subgroup of G here, not a subgroup of N.) I have been able to construct a number of examples of this, but I admit that my examples were all Abelian. I had hoped that this form might actually be easier to show, but to no avail, even in the Abelian case.

I am looking for ideas, perhaps even at most a general outline of a proof, but not a proof itself. I am having a number of severe difficulties with this section on cosets and I need to work out details on my own for now.

Thanks!

-Dan