Finite groups satisfying the independence property

TL;DR

This paper classifies finite groups with the independence property, showing they are all supersoluble, and provides a key theorem about the structure of almost simple groups related to subgroup conjugacy.

Contribution

It offers a complete classification of finite groups satisfying the independence property and proves they are supersoluble, advancing understanding of group generation properties.

Findings

All such groups are supersoluble.

Most almost simple groups contain an element with specific subgroup properties.

The classification includes a detailed analysis of subgroup conjugacy.

Abstract

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ 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 $H$ contain an element $s$ such that the maximal subgroups of $H$ containing $s$, but not containing the socle of $H$, are pairwise non-conjugate.