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**dwsmith** 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.