Nilpotent group that is finitely generated as a powered group for a set of primes

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: nilpotent group and group that is finitely generated as a powered group for a set of primes
View other group property conjunctions OR view all group properties

Definition

A group G is termed a nilpotent group that is finitely generated as a powered group for a set of primes if it satisfies the following equivalent conditions:

  1. G is a nilpotent group as well as a group that is finitely generated as a powered group for a set of primes. The latter means that there exists a subset \pi (possibly empty, possibly finite, possibly infinite, possibly including all primes) such that G is a \pi-powered group and there is a finite subset S of G such that there is no proper \pi-powered subgroup of G containing S.
  2. G is a nilpotent group and there is a set of primes \pi such that G is a pi-powered group, and further, the abelianization of G is finitely generated as a \mathbb{Z}[\pi^{-1}]-module.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group that is finitely generated as a module over the ring of integers localized at a set of primes |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group that is finitely generated as a powered group for a set of primes |FULL LIST, MORE INFO
nilpotent group |FULL LIST, MORE INFO