If a $p$-group $G$ acts on a finite set $S$, show that the number of fixed points of the action is congruent mod $p$ to the cardinality $|S|$ of the set $S$.

Let $F$ be the collection of fixed points. Since I know that the cardinality of the orbit $O$ for any given $s \in S$ equals $|G: Stab_{G}(s)|$, I can say all the fixed points, by definition, have orbits of cardinality 1. Because of the order of $G$, each non-trivial index must be a power of $p$. So, summing up cardinalities of the non-trivial orbits will give me some multiple of $p$. Is this correct or only partway there?


To remove this from the unanswered pool, the confirmation by anon in the comments seems to have resolved the question. So we add this as an answer.

"Yep, that's the idea."


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