p-group and group actions

Question

$G$ is a $p$-group, which means $|G|=p^n$ for $n\in \mathbb{Z^+}$.

Now,if $p$ does not divide $|S|$, for S is a set that G acts upon, how do I show that there exists $a\in S$ such that $G_a=G$

So how do I do this, I tried to find a relation with the order of $S$ but I really don't know how.

Answer 1

If you check my last comment under your question, from that lemma it follows that there must exist $\,a\in S\,$ s.t. $\,ga=a\;,\;\;\forall\,g\in G\,$ (why?).

For this very particular $\,a\in S\,$ it is certainly true that $\,G_a=G\,$ ...:)