Poincare's theorem about subgroups of finite index
I'm given this problem: let G be a group and A, B subgroups of finite order in G. Let . Prove that . (1) Using this, prove that (Poincare's theorem).
I can prove the theorem by defining is the set of right
cosets of D in G and so on for . f is one-to-one and therefor is a finite set. Furthermore, .
My question is: can (1) make the proof simpler or more straightforward than that given by me? (assuming it is correct). Because I've tried to use (1) to prove the theorem but I've failed. Any suggestion would be welcome. Thanks for reading.