Nilpotent p-group

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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

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