On the solvability of finite groups and the number of Sylow 2-subgroups

Abstract

Let G be a finite group. Denoted by n 2 (G) the number of Sylow 2-subgroups of G. In this paper, we prove if G is non-solvable and n 2 (G) is a power of a prime p, then p is a Fermat prime.Theorem. If G is non-solvable and the number of Sylow 2-subgroups of G is a power of a prime p, then p is a Fermat prime.