A group with exactly one noncommutator

Salem, Omar Hatem; Siniora, Daoud

doi:10.22108/ijgt.2025.146052.1975

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	A group with exactly one noncommutator
International Journal of Group Theory
دوره 15، شماره 3، آذر 2026، صفحه 161-167 اصل مقاله (407.37 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2025.146052.1975
نویسندگان
Omar Hatem Salem^* ؛ Daoud Siniora
Mathematics and Actuarial Science Department, The American University in Cairo, Cairo, Egypt
چکیده
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.
کلیدواژه‌ها
Commutator؛ Perfect groups؛ GAP

مراجع
[1] W. Benjamin Fite, On metabelian groups, Trans. Amer. Math. Soc., 3 no. 3 (1902) 331–353. [2] F. G. Frobenius, ¨Uber die Darstellung der endlichen gruppen durch lineare Substitutionen. Sitzungsberichte der K¨oniglich Preussischen Akademie der Wissenschaften zu Berlin, (1896) 994–1015. [3] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.15.1, 2025. https://www.gap-system.org [4] O. Hatem and D. Siniora, GAP code - A group with exactly one noncommutator. https://github.com/OmarHatemSalem/NonCommutators-GAP, 2025. [GitHub repository] [5] D. F. Holt and W. Plesken, Perfect groups, With an appendix and a supplement (on Microfiche) by W. Hanrath: Character tables of some factor groups of space groups, Oxford Math. Monogr, Oxford: Clarendon Press,1989. [6] I. M. Isaacs, Commutators and the commutator subgroup, Am. Math. Mon., 84 (1977) 720–722. [7] L.-C. Kappe and R. Fitzgerald Morse, On commutators in p-groups, J. Group Theory, 8 no. 4 (2005) 415–429. [8] E. I. Khukhro and V. D. Mazurov, Unsolved problems in group theory, The Kourovka Notebook, 2024. https://arxiv.org/abs/1401.0300 [9] M. W. Liebeck, E. A. O’Brien, A. Shalev and P. H. Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS), 12 no. 4 (2010) 939–1008. [10] I. D. MacDonald, Commutators and their products, Am. Math. Mon., 93 (1986) 440–444. [11] Ø. Ore, Some remarks on commutators, Proc. Am. Math. Soc., 2 (1951) 307–314. [12] J.-Pi. Serre, Repr´esentations lin´eaires des groupes finis. 5i`eme ´ed., corr. et augm. de nouveaux exercices, edition: 5i`eme ´ed., corr. et augm. de nouveaux exercices, Paris: Hermann, 1998. [13] S. V. Skresanov, On finite groups with exactly one noncommutator, 2025. https://arxiv.org/abs/2509.17587
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A group with exactly one noncommutator