Finite groups satisfying the independence property

Abstract

We say that a finite group [Formula: see text] satisfies the independence property if, for every pair of distinct elements [Formula: see text] and [Formula: see text] of [Formula: see text], either [Formula: see text] is contained in a minimal generating set for [Formula: see text] or one of [Formula: see text] and [Formula: see text] is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups [Formula: see text] contain an element [Formula: see text] such that the maximal subgroups of [Formula: see text] containing [Formula: see text], but not containing the socle of [Formula: see text], are pairwise non-conjugate.

2023 Ural Workshop on Group Theory and Combinatoric

“…For Γ P and Γ E , these are the groups in which every element has prime power order; they were investigated by many authors, a summary of results can be found, for example, in [6,Theorem 1.7]. For Γ P and Γ ′ I , and for Γ E and Γ ′ R , they are determined in a recent paper by S. Freedman et al [7]. The remaining case is open.…”