# Group with no infinite minimal generating set

From Groupprops

## Definition

A **group with no infinite minimal generating set** is a group in which any minimal generating set is a finite subset.

Note that such a group may or may not be a finitely generated group. If the group is a finitely generated group, then it does have minimal generating sets, all of which are finite. If the group is not finitely generated, it has no minimal generating set.

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

## Relation with other properties

### Stronger properties

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

Noetherian group | ||||

Artinian group | ||||

finitely generated group | ||||

group in which no subgroup has an infinite minimal generating set |