# Normalizer acts on stabilizer

1. Sep 22, 2009

### math_grl

1. The problem statement, all variables and given/known data
G acts transitively on S and let H be the stabilizer of s. Show that the normalizer of H, call it N, acts transitively on the fixpoints of H, call it F, where s is some element in S.

2. Relevant equations
Two different ways of showing this:
Either we show the orbit for any element in the fixpoints is the entire set of fixpoints
or we show that given any two fixpoints that there is an element in the normalizer such that it we can apply it to one of those elements and get the other one.

3. The attempt at a solution
There must be some trick or something small i'm missing.
Since G is tran. on S and F is a subset of S, then $$\forall s, t \in F, \exists g \in G$$ so that $$g s = t \Rightarrow s = g^{-1}t$$

Now $$ghs = ghg^{-1}t = h't = t$$ if and only if we can show that $$g \in N$$ since $$ghh'(gh)^{-1} = g(hh'h^{-1})g^{-1} \in N$$ whenever $$g \in N$$ and we have found our element in N, namely gh. I'm stuck, don't even know if i'm going in the right direction.

Last edited: Sep 22, 2009