A Group with Exactly One Noncommutator

TL;DR

This paper resolves an open problem in group theory by identifying the smallest finite groups where all but one element are commutators, using computational methods and structural analysis.

Contribution

It introduces an algorithmic approach combining theoretical properties and enumeration to find specific finite groups with a unique commutator property.

Findings

Identified two nonisomorphic groups of order 368,640 with the property

Established that 368,640 is the minimal order for such groups

Provided a structural description of one of the groups

Abstract

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.