Solvability of Finite Groups Determined by the Number of Sylow Subgroups

Asaraph Ansari; Abrar Ahmad

doi:https://doi.org/10.14445/22315373/IJMTT-V72I8P106

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 8 | Year 2026 | Article Id. IJMTT-V72I8P106 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I8P106

Solvability of Finite Groups Determined by the Number of Sylow Subgroups

Asaraph Ansari, Abrar Ahmad

Received	Revised	Accepted	Published
26 Jun 2026	27 Jul 2026	17 Aug 2026	30 Aug 2026

Citation :

Asaraph Ansari, Abrar Ahmad, "Solvability of Finite Groups Determined by the Number of Sylow Subgroups," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 8, pp. 48-53, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I8P106

Abstract

The relationship between the number of Sylow subgroups of a finite group and its solvability has long been a subject of interest in finite group theory. In this paper, we investigate the extent to which the solvability of a finite group is determined by restrictions on its Sylow numbers. Let 𝐺 be a finite group and, for each prime 𝑝, let 𝑛𝑝(𝐺) denote the number of Sylow 𝑝-subgroups of 𝐺. We establish several new solvability criteria expressed in terms of the set {𝑛𝑝(𝐺)∶ 𝑝| |𝐺| }. In particular, we prove that under certain numerical conditions on the Sylow numbers, the group 𝐺 must be solvable. Our results extend and refine earlier theorems of Luca, Navarro, Anabanti–Moretó–Zarrin, and Robati concerning the influence of Sylow numbers on the structure of finite groups.

Keywords

Finite groups, Sylow subgroups, Sylow numbers, solvable groups.

References

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