Abstract
A finite group G is called uniformly generated, if whenever there is a (strictly ascending) chain of subgroups 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.
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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.