The problem statement, all variables and given/known data Let H and K be subgroups of G of finite index such that [G] and [G:K] are relatively prime. Prove that G = HK. The attempt at a solution All I know is that [G intersect K] = [G] [G:K]. What would be nice is if [GK] = [G] [G:K] / [G intersect K], for then I would be done. Anywho, I must somehow show that [GK] = 1 or prove that G = HK directly. Any tips?