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

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
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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