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 < · · · < x 1 , x 2 , . . . , 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.

Finite Groups With Independent Generating Sets of Only Two Sizes

“…In particular, we obtain item (3) in Theorem 1.3. Let G be monolithic primitive with nonabelian socle N = S 1 × • • • × S n , with S S i for each 1 ≤ i ≤ n. The number [7] μ(G) = m(G) − m(G/N) has been investigated in [12]. The group G acts by conjugation on the set {S 1 , .…”

“…Let S := PSL 2 (7) and H := Aut(PSL 2 (7)), or let S := PSL 2 (9) and H ∈ {PΣL 2 (9), M 10 }. Consider the wreath product W := H Sym(n).…”

“…In view of this result, Apisa and Klopsch suggest a natural ‘classification problem’: given a nonnegative integer c , characterise all finite groups G which satisfy . The particular case has been recently highlighted by Glasby (see [7, Problem 2.3]).…”

“…https://doi.org/10.1017/S0004972723000126 Published online by Cambridge University Press[7] Independent generating sets…”