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    • Thread Starter

    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|HK|=\frac{|H||K|}{|H \cap K|}.

    where HK=\{hk:h\in H, k\in K\}

    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 \{H \cap K\} 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!

    Thanks in advance.

    (Original post by Kelvinator)
    I can prove the lemma when \{H \cap K\} 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 \phi : H \times K \rightarrow HK by \phi(h,k)=hk. 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.
    • Thread Starter

    Thanks for your help. I managed to prove it with the method you suggested.
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