---
tag: 098D2E7DEH
kind: statement
role: lemma
name: Fixed points of a p-group action
msc: [20D15, 20B05]
uses: [06DAH3VS6M, 08JF1K4DW1, 0S19CWFZM2, 0ZC7FYE0NZ]
sources:
  - url: https://en.wikipedia.org/wiki/P-group
    locator: Non-trivial center
  - url: https://en.wikipedia.org/wiki/Sylow_theorems
    locator: Proof of the Sylow theorems
---

## Statement

Let $p$ be a prime number, let $G$ be a finite [[p-group|06QYRTX9KQ]], and let $G$ [[act|0WCY42EV9V]] on a finite set $X$.
Let $X^G$ be the set of the [[fixed points|0RRWNMYRXC]] of the action.
Then $|X| \equiv |X^G| \pmod p$.

## Remarks

The [[orbits|0ZC7FYE0NZ]] partition $X$, and the orbits with one point are exactly the fixed points.
Each other orbit has a size that divides $|G|$ and is greater than $1$, so its size is divisible by $p$.
A consequence: if $p$ does not divide $|X|$, then the action has at least one fixed point.

## Explanations

### L1 human

The orbits of a $p$-group have sizes that are powers of $p$.
The orbits of size $1$ are the fixed points, and all other orbits have a size divisible by $p$.
So counting modulo $p$ sees only the fixed points.