Flexibility in generating sets of finite groups

TL;DR

This paper characterizes specific finite groups, namely certain affine groups, that have the property that any two non-generating elements can be extended to a minimal generating set, expanding understanding of group generation properties.

Contribution

It identifies and classifies the finite groups with the property that non-generating pairs can be extended to minimal generating sets, focusing on affine groups with particular structure.

Findings

Only specific affine groups have this property.

Elementary abelian groups extended by cyclic scalar actions are characterized.

Provides a complete classification of such finite groups.

Abstract

Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which finite groups, in addition, have the property that any two elements of G that do not generate a cyclic group can be extended to a generating set of minimal size. This note answers the question. The only such finite groups are very specific affine groups: elementary abelian groups extended by a cyclic group acting as scalars.