Group in which all minimal generating sets have the same size

From Groupprops
Jump to: navigation, search
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

Definition

Suppose G is a minimally generated group, i.e., it is a group with the property that it has a minimal generating set (note that any finite group is minimally generated). We say that G is a group in which all minimal generating sets have the same size if it is true that any two minimal generating sets of G have the same size (i.e., the same cardinality).

Note that this same size must therefore be the minimum size of generating set, and must also be the maximum size of minimal generating set. In fact, an alternative definition is that the minimum size of generating set must equal the maximum size of minimal generating set.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group of prime power order via Burnside's basis theorem