$p$-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
\[|X^P|\equiv |X| \pmod{p}.\]
Add to solve later 
