1. ## 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.

2. ## Re: Group Action

Originally Posted by dwsmith
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?

3. ## Re: Group Action

Originally Posted by Drexel28
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.

4. ## Re: Group Action

Originally Posted by dwsmith
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?

5. ## Re: Group Action

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