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.