## $p$-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

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

Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.

Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.

Then show that $N$ contains all $p$-Sylow subgroups of $G$.

Add to solve laterLet $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.

Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.

Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.

Let $p$ be a prime number.

Let $G$ be a non-abelian $p$-group.

Show that the index of the center of $G$ is divisible by $p^2$.

Read solution

Let $p$ be a prime number. Suppose that the order of each element of a finite group $G$ is a power of $p$. Then prove that $G$ is a $p$-group. Namely, the order of $G$ is a power of $p$.

Add to solve laterLet $G$ be a group of order $|G|=p^n$ for some $n \in \N$.

(Such a group is called a $p$*-group*.)

Show that the center $Z(G)$ of the group $G$ is not trivial.

Add to solve later