## $p$-Group Acting on a Finite Set and the Number of Fixed Points

## Problem 359

Let $P$ be a $p$-group acting on a finite set $X$.

Let

\[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \]

The prove that

\[|X^P|\equiv |X| \pmod{p}.\]

Let $P$ be a $p$-group acting on a finite set $X$.

Let

\[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \]

The prove that

\[|X^P|\equiv |X| \pmod{p}.\]