Groups with a base property analogous to that of vector spaces

2012/11/26 by Paul Apisa, Apisa, Paul, Benjamin Klopsch +1
Mathematics · #20F05 (Primary) 20D10 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20D10 #msc:20F05

paper · pdf · doi:10.48550/arxiv.1211.6137

10 pages

arxiv created 2012/11/26 · arxiv updated 2012/11/28

Abstract

A B-group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite B-groups is closed under taking quotients and that every finite B-group is solvable. Via a complete classification of Frattini-free finite B-groups we obtain a general structure theorem for finite B-groups. Applications include new proofs for the characterization of finite matroid groups and the classification of finite groups with the basis property.

Related

• Generating hypergraphs of finite groups
• Relative B-groups
• Notes on B-groups
• Flexibility in generating sets of finite groups
• Group bases for some solvable groups and semidirect products