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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?

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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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