Finite Groups with Certain Permutability Criteria

Abstract

Let G be a finite group. A subgroup H of G is said to be S-permutable in G if it
permutes with all Sylow subgroups of G. In this note we prove that if P, the Sylow
p-subgroup of G (p > 2), has a subgroup D such that 1 <|D|<|P| and all subgroups H of P with |H| = |D| are S-permutable in G, then G′ is p-nilpotent.