## $p$-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

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

Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.

Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, the $n$-dimensional vector space over $\F_p$. Therefore, $G_n$ acts on $\F_p^n$.

Let $e_n \in \F_p^n$ be the vector $(1,0, \dots,0)$.

(The so-called first standard basis vector in $\F_p^n$.)

Find the size of the $G_n$-orbit of $e_n$, and show that $\Stab_{G_n}(e_n)$ has order $|G_{n-1}|\cdot p^{n-1}$.

Conclude by induction that

\[|G_n|=p^{n^2}\prod_{i=1}^{n} \left(1-\frac{1}{p^i} \right).\]