Given a group G that acts on X transitively, P a Sylow p-subgroup of G and N the normalizer of P, define Y to be the subset of X whose points are all fixed by elements of P. How can I show that N acts transitively on Y?

Aside from regular manipulations, I've tried using the orbit-stabalizer theorem to show there's only one orbit to no avail. I can't seem to use the Sylow p-subgroup assumption. Any hints?

  • $\begingroup$ I am thinking. Can you tell me the reference/book where it is from? $\endgroup$ – Bhaskar Vashishth Dec 10 '14 at 9:58

This is a good exercise in applying the conjugacy part of Sylow's Theorem. Here is an outline of the proof.

Let $x,y \in Y$. There exists $g \in G$ with $g(x)=y$. Now $P,gPg^{-1}$ are Sylow $p$-subgroups of $G_y$, so there exists $h \in G_y$ with $hgPg^{-1}h^{-1} = P$, and now $hg(x)=y$ with $hg \in N$.


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