---
tag: 06DAH3VS6M
kind: definition
name: P-group
msc: [20D15]
defines: 06QYRTX9KQ
uses: [0JPYSD5E0B]
sources:
  - url: https://en.wikipedia.org/wiki/P-group
    locator: lead section
---

## Definition

Let $p$ be a prime number.
A [[group|G5S1M9]] $G$ is a $p$-group when, for each $g \in G$, the [[order|0WE2NHZJ7B]] of $g$ is $p^n$ for some integer $n \geq 0$.

## Remarks

A finite group is a $p$-group if and only if $|G| = p^k$ for some $k \geq 0$.
One direction uses [[Lagrange's theorem|3C53TD]], and the other direction uses [[Cauchy's theorem|0JM7PA288B]].
The [[dihedral group|0JZ27DMJ5D]] $D_4$ of order $8$ is a $2$-group that is not abelian.
The trivial group is a $p$-group for each prime $p$.

## Explanations

### L1 human

In a $p$-group, each element comes back to the identity after a power of $p$ steps.
For finite groups this means that the number of elements is a power of $p$.
These groups are the building blocks of the Sylow theory.