Finite groups with the minimal generating set exchange property

TL;DR

This paper classifies finite groups based on a property related to minimal generating sets and their element exchangeability, providing a comprehensive understanding of their structure and proposing a conjecture about generating sets relative to maximal subgroups.

Contribution

It offers a complete classification of finite groups with the minimal generating set exchange property and proves a related conjecture for certain cases.

Findings

Classified finite groups with the exchange property.

Proved existence of minimal generating sets with most elements in a maximal subgroup.

Conjectured and partially proved that only one element outside the subgroup is needed.

Abstract

Let $d(G)$ be the smallest cardinality of a generating set of a finite group $G.$ We give a complete classification of the finite groups with the property that, whenever $ \langle x_1, \dots, x_{d(G)} \rangle = \langle y_1, \dots, y_{d(G)} \rangle = G$, for any $1 \leq i \leq d(G)$ there exists $1 \leq j \leq d(G)$ such that $\langle x_1, \dots, x_{i-1}, y_j, x_{i+1}, \dots, x_{d(G)} \rangle = G.$ We also prove that for every finite group $G$ and every maximal subgroup $M$ of $G$, there exists a generating set for $G$ of minimal size in which at least $d(G)-2$ elements belong to $M$. We conjecture that the stronger statement holds, that there exists a generating set of size $d(G)$ in which only one element does not belong to $M$, and we prove this conjecture for some suitable choices of $M$.