On the additive group of a finite skew brace whose multiplicative group is a product of two non-abelian simple groups

Articles in Press

Document Type : Research Paper

Author

Dipartimento di Matematica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy

Abstract

We study finite skew braces whose multiplicative group is the direct product of two finite non-abelian simple groups. More precisely, we show that if B is a finite skew brace with $(B,\cdot)\cong S\times T,$ where S and T are finite non-abelian simple groups, then the additive group (B,+) cannot be supersolvable. This result contributes to the study of the restrictions imposed by the multiplicative group on the additive group of a skew brace, a central question related to the Byott--Vendramin conjecture. The proof combines structural properties of supersolvable groups, the existence of characteristic Hall subgroups, and classification results on finite simple groups admitting subgroups of 2-power index. These tools reduce the possible multiplicative groups to groups of projective linear type and force the occurrence of a section isomorphic to PSL2(7). The remaining cases are excluded through an analysis of suitable Sylow subgroups and the induced lambda action.

Keywords

Main Subjects

References

[1] N. P. Byott, Hopf-Galois structures on field extensions with simple Galois groups, Bull. London Math. Soc., 36 no. 1 (2004) 23–29.
[2] N. P. Byott, Solubility criteria for Hopf-Galois structures, New York J. Math., 21 (2015) 883–903.
[3] N. P. Byott, On insoluble transitive subgroups in the holomorph of a finite soluble group, J. Algebra, 638 (2024) 1–31.
[4] M. Damele, Finite skew braces whose additive group is a Z-group, Arch. Math., to appear; arXiv:2603.22980.
[5] M. Damele and A. Loi, Structural and rigidity properties of Lie skew braces, J. Algebra, 695 (2026) 356–383.
[6] W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math., 13 (1963) 775–1029.
[7] The GAP Group, GAP – Groups, Algorithms, and Programming, https://www.gap-system.org.
[8] I. Gorshkov and T. Nasybullov, Finite skew braces with solvable additive group, J. Algebra, 574 (2021) 172–183.
[9] L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comp., 86 no. 307 (2017) 2519–2534.
[10] B. Huppert, Endliche Gruppen, I. (German) Die Grundlehren der mathematischen Wissenschaften, 134, Springer-Verlag, Berlin-New York, 1967.
[11] I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008.
[12] A. Koch and P. J. Truman, Opposite skew left braces and applications, J. Algebra, 546 (2020) 218–235.
[13] B. Li, T. Nasybullov and V. Zadvornov, Comparison of addition and multiplication in a skew brace, preprint, 2025; arXiv:2511.22322.
[14] T. Nasybullov, Connections between properties of the additive and the multiplicative groups of a two-sided skew brace, J. Algebra, 540 (2019) 156–167.
[15] R. Schoof, Catalan’s conjecture, Universitext. Springer-Verlag London, Ltd., London, 2008.
[16] A. Smoktunowicz and L. Vendramin, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Combin. Algebra, 2 no. 1 (2018) 47–86.
[17] C. S. Y. Tsang, Hopf–Galois structures on finite extensions with quasisimple Galois group, Bull. London Math. Soc., 53 no. 1 (2021) 148–160.
[18] C. Tsang and C. Qin, On the solvability of regular subgroups in the holomorph of a finite solvable group, Internat. J. Algebra Comput., 30 no. 2 (2020) 253–265.
[19] L. Vendramin, Problems on skew left braces, Adv. Group Theory Appl., 7 (2019) 15–37.
International Journal of Group Theory

Articles in Press, Accepted Manuscript
Available Online from 06 August 2026

Files

History

  • Receive Date: 01 June 2026
  • Revise Date: 20 July 2026
  • Accept Date: 06 August 2026
  • Published Online: 06 August 2026

Share

How to cite

Statistics