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Groupprops β

Group with no infinite minimal generating set

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 properties
VIEW 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