Quasi-central elements and p-nilpotence of finite groups

Let G be a finite group and let P be a Sylow p-subgroup of G. An element x of G is called quasi-central in G if < x >< y > = < y >< x > for each y is an element of G. In this paper, it is proved that G is p-nilpotent if and only if N-G(P) is p-nilpotent and, for all x is an element of G\N-G(P), one of the following conditions holds: (a) every element of P boolean AND P-x boolean AND G(Np) of order p or 4 is quasi-central in P; (b) every element of P boolean AND P-x boolean AND G(Np) of order p is quasi-central in P and, if p = 2, P P-x boolean AND G(Np) is quaternion-free; (c) every element of P boolean AND P-x boolean AND G(Np) of order p is quasi-central in P and, if p = 2, [Omega(2)(P boolean AND P-x boolean AND GNP), <= Z(P boolean AND G(Np)); (d) every element of P boolean AND G(Np) of order p is quasi-central in P and, if p = 2, [Omega(2)(P boolean AND P-x boolean AND G(Np)), P] <= Omega(1)(P boolean AND G(Np)); (e) vertical bar Omega(1)(P boolean AND P-x boolean AND G(Np))vertical bar <= p(p-1) and, if p = 2, P boolean AND P-x boolean AND G(Np) is quaternion-free; (f) vertical bar Omega(P boolean AND P-x boolean AND < G(Np))vertical bar <= p(p-1). That will extend and improve some known related results.