Classifying uniformly generated groups

Abstract

A finite group G is called uniformly generated, if whenever there is a (strictly ascending) chain of subgroups $1<〈x_1〉<〈x_1,x_2〉<\cdots <〈x_1,x_2,\dots ,x_d〉=G,$ then d is the minimal number of generators of G. Our main result classifies the uniformly generated groups without using the simple group classification. These groups are related to finite projective geometries by a result of Iwasawa on subgroup lattices.

Keywords:

• Length
• depth
• irredundant
• subgroup chain

Mathematics Subject Classification:

• 20E15
• 14N20

Acknowledgments

The problem of classifying uniformly generated groups was posed by the author at the 2018 CMSC Annual Research Retreat, and solved promptly. I thank the CMSC for hosting the Retreat, and Scott Harper for his helpful comments. Finally, I thank the referee for suggesting improvements to this note.

Additional information

Funding

I acknowledge the support of the Australian Research Council Discovery Grants DP160102323 and DP190100450.