(Actually, this is more of a follow up question rather than an answer.)  

Wouldn't it make more sense if the complement of a subgroup $H \leq G$ were defined to be a subgroup $K\leq G$ such that $H \cap K = 1$ and $\langle H, K \rangle = G$?  That is, shouldn't we allow for the possibility that the set $HK$, consisting of all products $h k$ ($h\in H, k\in K$), is not a group? (Yet we may still have $HK \subsetneq \langle H, K \rangle = G$ in this situation.)

..or does everyone take $HK$ to mean $\langle H, K \rangle$ in this context?

Thanks,
William