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Math Help - Group Action

  1. #1
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    Group Action

    Let G be a finite group acting on the set S. Suppose H\trianglelefteq G so that for any s_1,s_2\in S there is a unique h\in H so that hs_1=s_2. For each s\in S, let G_s=\{g\in G: gs=s\}.

    Prove:

    G=G_sH \ \text{and} \ G_s\cap H=\{e\}

    I think about trying to use the order since if |G|=|G_sH|=\frac{|G_s||H|}{|H\cap G_s|} but other than that I don't know what I need to do.
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    MHF Contributor Drexel28's Avatar
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    Re: Group Action

    Quote Originally Posted by dwsmith View Post
    Let G be a finite group acting on the set S. Suppose H\trianglelefteq G so that for any s_1,s_2\in S there is a unique h\in H so that hs_1=s_2. For each s\in S, let G_s=\{g\in G: gs=s\}.

    Prove:

    G=G_sH \ \text{and} \ G_s\cap H=\{e\}

    I think about trying to use the order since if |G|=|G_sH|=\frac{|G_s||H|}{|H\cap G_s|} but other than that I don't know what I need to do.
    You made the correct observation. If we can prove that \text{stab}(s)\cap H=\{e\} then |\text{stab}(s)H|=|G| and so \text{stab}(s)H=G. Now, suppose for a second that h\in \text{stab}(s)\cap H then you know hs=s but you know that there is a UNIQUE h which does this. What could this h be?
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    Re: Group Action

    Quote Originally Posted by Drexel28 View Post
    You made the correct observation. If we can prove that \text{stab}(s)\cap H=\{e\} then |\text{stab}(s)H|=|G| and so \text{stab}(s)H=G. Now, suppose for a second that h\in \text{stab}(s)\cap H then you know hs=s but you know that there is a UNIQUE h which does this. What could this h be?
    You have hs=s which is true if h\in G_s but the unique h is for hs_1=s_2 where s_1 \ \text{and} \ s_2 aren't necessarily the same.
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    MHF Contributor Drexel28's Avatar
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    Re: Group Action

    Quote Originally Posted by dwsmith View Post
    You have hs=s which is true if h\in G_s but the unique h is for hs_1=s_2 where s_1 \ \text{and} \ s_2 aren't necessarily the same.
    You know that FOR ANY two elements of S there exists a UNIQUE element of H sending one to the other. What always fixes s, and how does that help us?
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    Re: Group Action

    but IF s_1 = s_2 = s, we know therefore that the ONLY element of H in Stab(s) is e.
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