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# Tricky group theory proof watch

1. Hi, I am trying to prove the following result:

Lemma:Let G be a group and let H,K be finite subgroups of G. Then.

where

I know |HK| corresponds to the number of ways I can do h*k (this corresponds to |H|*|K|) subtract all the repeated h*k. I can prove the lemma when only has one element (the identity) but not when H and K share more elements. I have a feeling I am missing something really obvious but I can't figure it out!

2. (Original post by Kelvinator)
I can prove the lemma when only has one element (the identity) but not when H and K share more elements.
That's a good start.

There might be better ways of doing this, but my first thought is to do define a map by . You can then argue that this map is surjective, and that for every element in HK, there are exactly |HnK| elements in HxK which are mapped to it by phi - see if you can prove this, and see if you can finish the proof off from here.
3. Thanks for your help. I managed to prove it with the method you suggested.

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