Prime power Sylow numbers

Document Type : Research Paper

Author

Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran

Abstract

In this paper, we study finite groups with a prime-power number of Sylow \(p\)-subgroups. Motivated by the work of Yang et al.~(2022), who characterized non-solvable groups with a prime-power number of Sylow $2$-subgroups, we investigate the corresponding problem for odd primes. We prove that if a finite group \(G\) has a non-abelian composition factor whose order is divisible by an odd prime \(p\), and the number of Sylow \(p\)-subgroups is a prime power, then \(p\) must be a Mersenne prime and \(n_p(G)=2^k\) for some integer \(k\ge 2\).

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References

[1] Anabanti, C.S., Moreto, A., & Zarrin, M. (2020). Influence of the number of Sylow subgroups on solvability of  nite groups. C. R. Math, 358, 1227-1230. https://doi.org/10.5802/crmath.146
[2] B. Brewster, B., & Ward, M.B. (1995). Groups 2-transitive on a set of their Sylow subgroups. Bull. Austral. Math. Soc, 52, 117{136. https://doi.org/10.1017/S0004972700014507
[3] Guralnick, R.M. (1983). Subgroups of prime power index in a simple group. J. Algebra, 81(2), 304{311. https://doi.org/10.1016/0021-8693(83)90190-4
[4] Hall, M. (1967). On the number of Sylow subgroups in a  finite group. J. Algebra, 7(3), 363{371. https://doi.org/10.1016/0021-8693(67)90076-2
[5] Jones, G. A., & Sezer, S. Permutation groups of prime power degree and p-complements. arXiv:2402.13672v2.
[6] Yang, R., Yang, Y., & Shen, R. (2022). On the solvability of  finite groups and the number of Sylow 2-subgroups. Eur. J. Math. Appl, 2(3) (2022), 1-5. https://doi.org/10.28919/ejma.2022.2.3
[7] The GAP Group, GAP-Groups, Algorithms, and Programming. Version 4.13.1 (2024). https://www.gap-system.org 

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History

  • Receive Date: 25 November 2025
  • Revise Date: 08 February 2026
  • Accept Date: 12 April 2026

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