A note on solvability of finite groups

Abstract

Let G be a finite group. A subgroup H of G is said to be c-normal in G if there exists a normal subgroup K of G such that G = HK and H K -<H_G, where H_G is the largest normal subgroup of G contained in H. In this note we prove that if every Sylow subgroup P of G has a subgroup D such that 1 <|D|<|P| and all subgroups H of P with |H|=|D|are c-normal (S-permutable) in G, then G is solvable. This results improve and extend classical and recent results in the literature.