1. The problem statement, all variables and given/known data Let G be a group and let H,K be subgroups of G. Assume that G is finite and that the indices |G| and |G:K| are relatively prime. Show that G=HK. Hint: Show that |G(intersect)K| is divisible by both |G| and |G:K| and then use the counting principle for |HK|. 3. The attempt at a solution First off, why do the indices have to be relatively prime? I don't know how to show that |G(intersect)K| is divisible by both |G| and |G:K|, but I do know that if I assume those, I know how to use the counting principle because ultimately it will come down to saying that |HK|=c|G| for some multiple c, and c must = 1 otherwise it says that for c>1, |HK|>|G| and that is not possible. EDIT: Is the intersection of the left coset of H and the left coset of K disjoint? Since they are both equivalence classes they would have to either be disjoint or equal, no? So then |G(intersect)K| would consist of both xH and xK for some x in G.....?