$G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$?
I tried to ask this question yesterday but did not word it right, so the one who helped me claim that this part is clearly true and I really do not understand why, and I couldn't get a straight answer for the past day.
Any of you are welcome to vote to delete this question (after some discussion with me about the problem I hope), but I just really want to know why it is "clearly true, because I still couldn't reason it right.