Abelian group that is finitely generated as a module over the ring of integers localized at 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: 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
finitely generated abelian group |FULL LIST, MORE INFO