# Group of prime power order

From Groupprops

(Redirected from Finite p-group)

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter

View other prime-parametrized group properties | View other group properties

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)

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## Contents

## Definition

### Symbol-free definition

A **group of prime power order** is defined in the following equivalent ways:

- It is a finite group whose order is a power of a prime.
- It is a finite group that is also a p-group for some prime : the order of every element is a power of that same prime

### Equivalence of definitions

`For full proof, refer: Equivalence of definitions of group of prime power order`

## Relation with other properties

### Weaker properties

- Nilpotent group:
`For full proof, refer: Prime power order implies nilpotent` - Solvable group
- Group whose order has at most two prime factors

## Specific information

See groups of prime power order for more specific information.