Results 1 to 5 of 5

Thread: Group Action

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    10

    Group Action

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

    Prove:

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

    I think about trying to use the order since if $\displaystyle |G|=|G_sH|=\frac{|G_s||H|}{|H\cap G_s|}$ but other than that I don't know what I need to do.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22

    Re: Group Action

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

    Prove:

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

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

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    10

    Re: Group Action

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

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22

    Re: Group Action

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

  5. #5
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,546
    Thanks
    842

    Re: Group Action

    but IF $\displaystyle s_1 = s_2 = s$, we know therefore that the ONLY element of H in Stab(s) is e.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Group action G on itself
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Oct 21st 2011, 03:40 PM
  2. Kernel of a group action
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Oct 19th 2011, 03:02 AM
  3. group action
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Aug 26th 2010, 12:07 PM
  4. Group action
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Apr 15th 2010, 12:49 PM
  5. Group action
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 15th 2009, 12:58 PM

Search Tags


/mathhelpforum @mathhelpforum