# Nilpotent p-group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: p-group and nilpotent group

View other group property conjunctions OR view all group properties

## Contents

## Definition

Let be a prime. A **nilpotent p-group** is a group satisfying the following equivalent conditions:

- It is a -group (see p-group -- every element has order a power of ) that is also a nilpotent group.
- It is a a nilpotent group in which every finitely generated subgroup is a finite p-group.

## Facts

- Every finite -group is nilpotent -- see prime power order implies nilpotent.
- There exist infinite non-nilpotent -groups. See p-group not implies nilpotent.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group of prime power order | finite -group | prime power order implies nilpotent | ||

abelian p-group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

p-group | p-group not implies nilpotent | |||

hypercentral p-group | ||||

solvable p-group | ||||

periodic nilpotent group | ||||

locally finite group | ||||

periodic group |