Finite soluble groups satisfying the replacement property

TL;DR

This paper studies finite soluble groups that satisfy a specific replacement property, showing how elements can replace generators in maximal irredundant generating sets while still generating the entire group.

Contribution

It characterizes finite soluble groups with the replacement property, providing new insights into their generating set structures.

Findings

Finite soluble groups satisfying the replacement property are characterized.

The replacement property allows elements to replace generators in maximal irredundant sets.

Results contribute to understanding the structure of generating sets in soluble groups.

Abstract

We investigate the finite soluble groups $G$ with the following property (replacement property): for every irredundant generating set $\{g_1,\dots,g_m\}$ of maximal size and for any $1\neq g\in G$ there exists an $i\in \{1,\dots,m\}$ so that $\langle g_1,\dots,g_{i-1},g,g_{i+1},\dots,g_m\rangle=G.$