The overall problem is this:
I have tried various and sundry methods to work with this, to little result. I have managed to prove that is a subgroup of G, but that was from a tactic that I eventually had to consider unfruitful.If H < N and H, N, and K are subgroups of G, then .
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: . (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.