On finite groups with exactly one noncommutator

2025/09/22 by Saveliy V. Skresanov, Skresanov, Saveliy V.
Engineering · Mathematics · #20F12 (Primary) 20-08 (Secondary) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2509.17587

openalex publication_date 2025/09/22 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

An element x of a group G is a commutator if it can be expressed in the form x = a^-1b^-1ab for some a, b ∈ G. In 2010 MacHale posed the following problem in the Kourovka notebook: does there exist a finite group G, with |G| > 2, such that there is exactly one element of G which is not a commutator? We answer this question in the affirmative and provide an infinite series of such groups, the smallest group in our construction having size 16609443840.

Related

• Commutators of elements of coprime orders in finite groups
• Groups with Finiteness Conditions on commutator subgroups
• Finite-by-nilpotent groups and a variation of the BFC-theorem
• Commutators in finite p-groups with 2-generator derived subgroup
• Commutativity degree, its generalizations, and classification of finite groups