Finite groups in which there are only two possible cardinalities for an independent generating set

TL;DR

This paper investigates finite groups where the sizes of independent generating sets are restricted to only two possible values, addressing a problem proposed by Peter Glasby.

Contribution

It characterizes finite groups with exactly two possible sizes for independent generating sets, advancing understanding of their structural properties.

Findings

Finite groups with two independent generating set sizes are classified.

The structure of such groups is explicitly described.

New insights into the relationship between group structure and generating set properties.

Abstract

A generating set $S$ for a group $G$ is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in $G$, for all $s \in S.$ In this paper, we study a problem proposed by Peter Glasby: we investigate finite groups, where there are only two possible cardinalities for the independet generating sets.