A group with exactly one noncommutator

Document Type : Research Paper

Authors

Mathematics and Actuarial Science Department, The American University in Cairo, Cairo, Egypt

10.22108/ijgt.2025.146052.1975

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. 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 Holt and Plesken’s list 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. We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this is the smallest order of such a group. 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.

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References

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International Journal of Group Theory
Volume 15, Issue 3 - Serial Number 3
September 2026
Pages 161-167

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History

  • Receive Date: 30 July 2025
  • Revise Date: 24 November 2025
  • Accept Date: 24 November 2025
  • Published Online: 26 November 2025

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