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.

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