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

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A group is termed a **group that is finitely generated as a powered group for a set of primes** if there exists a subset (possibly empty, possibly finite, possibly infinite, possibly including all primes) such that is a -powered group and there is a finite subset of such that is a generating set in the -powered sense, i.e., there is no proper -powered subgroup of containing .

## 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 | the group is also an abelian group | Nilpotent group that is finitely generated as a powered group for a set of primes|FULL LIST, MORE INFO | ||

nilpotent group that is finitely generated as a powered group for a set of primes | |FULL LIST, MORE INFO |