Finite groups with the minimal generating set exchange property

2025/06/02 by Andrea Lucchini, Lucchini, Andrea, Patricia Medina Capilla +1
Computer Science · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2506.01638

openalex publication_date 2025/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ⟨ x₁, …, x_d(G) ⟩ = ⟨ y₁, …, y_d(G) ⟩ = G, for any 1 ≤ i ≤ d(G) there exists 1 ≤ j ≤ d(G) such that ⟨ x₁, …, x_i-1, y_j, x_i+1, …, x_d(G) ⟩ = 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.

Citations

• Generating hypergraphs of finite groups
• On the connectivity of the non-generating graph
• The spread of a finite group
• Generation of finite simple groups by an involution and an element of prime order

Related

• Flexibility in generating sets of finite groups
• The Minimum Generating Set Problem
• Minimal invariable generating sets
• Finite groups in which there are only two possible cardinalities for an independent generating set
• The maximal size of a minimal generating set