Abelian group that is finitely generated as a module over the ring of integers localized at a set of primes
From Groupprops
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 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 of the set of prime numbers such that is a finitely generated as a module over the ring . Another way of putting it is that there is a finite subset of such that the -powered subgroup generated by is the whole group .
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | No | It is possible to have a group that is finitely generated as a module over for some prime set , and a subgroup of that is not finitely generated as a module over any prime set . | |
quotient-closed group property | No | It is possible to have a group that is finitely generated as a module over for some prime set , and a subgroup of such that the quotient group is not finitely generated as a module over any prime set . |
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 |