Nilpotent p-group

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

Definition

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

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

Facts

• Every finite $p$-group is nilpotent -- see prime power order implies nilpotent.
• There exist infinite non-nilpotent $p$-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 $p$-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