Solvability of Finite Groups Determined by the Number of Sylow Subgroups

Source 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.