On p-nilpotence of finite groups

Abstract

All groups considered in this paper will be finite. A 2-group is called quaternion-free if it has no section isomorphic to the quaternion group of order 8. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω 1(P) . Zhang [Proc. Amer. Math. Soc. 98 (4) (1986) 579] proved that if P is a Sylow p-subgroup of G, Ω 1(P)⩽Z(P) and N G ( Z( P)) is p-nilpotent, then G is p-nilpotent, i.e., G has a normal Hall p′-subgroup. Recently, Ballester-Bolinches and Guo [J. Algebra 228 (2000) 491] proved that if P is a Sylow 2-subgroup G, P is quaternion-free, Ω 1(P∩G′)⩽Z(P) and N G ( P) is 2-nilpotent, then G is 2-nilpotent. Bannuscher and Tiedt [Ann. Univ. Sci. Budapest 37 (1994) 9] proved that if p>2, P is a Sylow p-subgroup of G, |Ω 1(P∩P x)|⩽p p−1 for all x∈ G⧹ N G ( P) and N G ( P) is p-nilpotent, then G is p-nilpotent. The object of this paper is to improve and extend these results.