On generalized quaternion Sylow $ 2 $-subgroups and $ 2 $-nilpotence of finite groups

Fawaz Aseeri; Julian Kaspczyk; Fawaz Aseeri; Julian Kaspczyk

doi:10.3934/math.2026606

AIMS Mathematics

2026, Volume 11, Issue 5: 14757-14762. doi: 10.3934/math.2026606

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Research article

On generalized quaternion Sylow $ 2 $-subgroups and $ 2 $-nilpotence of finite groups

Fawaz Aseeri ^{1
,
,},
Julian Kaspczyk ²

1.
Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia
2.
Berlin, Germany

Received: 11 March 2026 Revised: 29 April 2026 Accepted: 11 May 2026 Published: 26 May 2026
MSC : 20D10, 20D15, 20D20

In 2020, Mousavi proved that a finite group $ G $ with a generalized quaternion Sylow $ 2 $-subgroup $ S $ is $ 2 $-nilpotent if $ 3 \nmid |G| $ or if $ G $ is solvable and $ |S| > 16 $. In this note, we generalized the result of Mousavi and provided a simpler proof. In detail, we showed that a finite group with a generalized quaternion Sylow $ 2 $-subgroup is $ 2 $-nilpotent if, and only if, it is $ \mathrm{SL_2(3)} $-free and that, for a finite group with a generalized quaterion Sylow $ 2 $-subgroup of order strictly greater than $ 16 $, the properties of being $ 2 $-nilpotent, solvable, and $ 2 $-constrained are equivalent.
- finite group,
- $ p $-nilpotent,
- Sylow 2-subgroup,
- quaternion group,
- $ p $-constrained
Citation: Fawaz Aseeri, Julian Kaspczyk. On generalized quaternion Sylow $ 2 $-subgroups and $ 2 $-nilpotence of finite groups[J]. AIMS Mathematics, 2026, 11(5): 14757-14762. doi: 10.3934/math.2026606

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Abstract

In 2020, Mousavi proved that a finite group $ G $ with a generalized quaternion Sylow $ 2 $-subgroup $ S $ is $ 2 $-nilpotent if $ 3 \nmid |G| $ or if $ G $ is solvable and $ |S| > 16 $. In this note, we generalized the result of Mousavi and provided a simpler proof. In detail, we showed that a finite group with a generalized quaternion Sylow $ 2 $-subgroup is $ 2 $-nilpotent if, and only if, it is $ \mathrm{SL_2(3)} $-free and that, for a finite group with a generalized quaterion Sylow $ 2 $-subgroup of order strictly greater than $ 16 $, the properties of being $ 2 $-nilpotent, solvable, and $ 2 $-constrained are equivalent.

References

[1]	M. Asaad, M. Ramadan, A. Heliel, Influence of weakly $\mathcal{H}$-embedded subgroups on the structure of finite groups, Publ. Math. Debrecen, 91 (2017), 503–513. https://doi.org/10.5486/PMD.2017.7842 doi: 10.5486/PMD.2017.7842
[2]	A. Ballester-Bolinches, X. Y. Guo, Y. M. Li, N. Su, On finite $p$-nilpotent groups, Monatsh. Math., 181 (2016), 63–70. https://doi.org/10.1007/s00605-015-0803-y doi: 10.1007/s00605-015-0803-y
[3]	D. A. Craven, The theory of fusion systems, Cambridge: Cambridge University Press, 2011. https://doi.org/10.1017/CBO9780511794506
[4]	Y. X. Gao, X. H. Li, D. L. Lei, $p$-Sylowizers and $p$-nilpotency of finite groups, Indian J. Pure Appl. Math., 56 (2025), 1734–1740. https://doi.org/10.1007/s13226-024-00674-5 doi: 10.1007/s13226-024-00674-5
[5]	The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.15.1, 2025. Available from: https://www.gap-system.org.
[6]	W. Gaschütz, Sylowisatoren, Math. Z., 122 (1971), 319–320. https://doi.org/10.1007/BF01110166 doi: 10.1007/BF01110166
[7]	D. Gorenstein, Finite groups, 2Eds., New York: AMS Chelsea Publishing, 1980.
[8]	Y. H. Guo, I. M. Isaacs, Conditions on $p$-subgroups implying $p$-nilpotence or $p$-supersolvability, Arch. Math., 105 (2015), 215–222. https://doi.org/10.1007/s00013-015-0803-0 doi: 10.1007/s00013-015-0803-0
[9]	B. Huppert, Endliche gruppen I, Berlin: Springer, 1967. https://doi.org/10.1007/978-3-642-64981-3
[10]	B. Huppert, Character theory of finite groups, New York: De Gruyter, 1998. https://doi.org/10.1515/9783110809237
[11]	I. M. Isaacs, Finite group theory, Providence: American Mathematical Society, 2008. https://doi.org/10.1090/gsm/092
[12]	H. Kurzweil, B. Stellmacher, The theory of finite groups: An introduction, New York: Springer, 2004. https://doi.org/10.1007/b97433
[13]	Y. M. Li, N. Su, Y. M. Wang, A generalization of Burnside's $p$-nilpotency criterion, J. Group Theory, 20 (2017), 185–192. https://doi.org/10.1515/jgth-2016-0028 doi: 10.1515/jgth-2016-0028
[14]	H. Mousavi, Finite groups with quaternion Sylow subgroup, C. R. Math., 358 (2020), 1097–1099. https://doi.org/10.5802/crmath.131 doi: 10.5802/crmath.131
[15]	H. R. Yu, Y. Guan, X. W. Xu, A sufficient and necessary condition for $p$-nilpotence of a finite group, J. Algebra Appl., 25 (2026), 2550321. https://doi.org/10.1142/S0219498825503219 doi: 10.1142/S0219498825503219

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