A solvability criterion for finite groups related to the number of Sylow subgroups

Abstract

Let G be a finite group and let $\pi (G)$ be the set of primes dividing the order of G. For each $p\in \pi (G),$ the Sylow theorems state that the number of Sylow p-subgroups of G is equal to kp + 1 for some non-negative integer k. In this article, we characterize non-solvable groups G containing at most $p^2+1$ Sylow p-subgroups for each $p\in \pi (G).$ In particular, we show that each finite group G containing at most $(p-1)p+1$ Sylow p-subgroups for each $p\in \pi (G)$ is solvable.

Keywords:

• Finite groups
• Sylow subgroups
• simple groups

MATHEMATICS SUBJECT CLASSIFICATION:

• 20D20; 20D05

Acknowledgment

The author would like to thank Dr. Farrokh Shirjian for the insightful discussions.