Quasi-central elements and $p$-nilpotence of finite groups

Abstract

Let G be a ¯nite group and let P be a Sylow p-subgroup of G. An element x of G is called quasi-central in G if hxihyi = hyihxi for each y 2 G. In this paper, it is proved that G is p-nilpotent if and only if NG(P) is p-nilpotent and, for all x 2 GnNG(P), one of the following conditions holds: (a) every element of P Px GNp of order p or 4 is quasi-central in P; (b) every element of P Px GNp of order p is quasi- central in P and, if p = 2, P Px GNp is quaternion-free; (c) every element of P Px \GNp of order p is quasi-central in P and, if p = 2, [­2(P Px \GNp ); P] · Z(P GNp ); (d) every element of P GNp of order p is quasi-central in P and, if p = 2, [­2(P Px GNp ); P] · ­1(P GNp ); (e) j­1(P Px GNp )j · ppi1 and, if p = 2, P Px GNp is quaternion-free; (f) j­(P Px GNp )j · ppi1. That will extend and improve some known related results.