# Abelian group that is finitely generated as a module over the ring of integers localized at a set of primes

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: abelian 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 abelian group that is finitely generated as a module over the ring of integers localized at a set of primes if it satisfies the following: there is a (possibly empty, possibly finite, possibly infinite) subset $\pi$ of the set of prime numbers such that $G$ is a finitely generated as a module over the ring $\mathbb{Z}[\pi^{-1}]$. Another way of putting it is that there is a finite subset $S$ of $G$ such that the $\pi$-powered subgroup generated by $S$ is the whole group $G$.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property No It is possible to have a group $G$ that is finitely generated as a module over $\mathbb{Z}[\pi^{-1}]$ for some prime set $\pi$, and a subgroup $H$ of $G$ that is not finitely generated as a module over any prime set $\pi$.
quotient-closed group property No It is possible to have a group $G$ that is finitely generated as a module over $\mathbb{Z}[\pi^{-1}]$ for some prime set $\pi$, and a subgroup $H$ of $G$ such that the quotient group $G/H$ is not finitely generated as a module over any prime set $\pi$.

## Relation with other properties

### Stronger properties

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