On \(p\)-nilpotence of finite groups. (Q1879638)

A finite group \(G\) is said to be \(p\)-nilpotent (where \(p\) is a prime) if it has a normal Hall \(p'\)-subgroup, that is, \(O_{p'p}(G)=G\). Obviously, every finite nilpotent group is \(p\)-nilpotent and conversely a finite group which is \(p\)-nilpotent for all \(p\) is nilpotent. Many authors have established criteria for \(p\)-nilpotence. Classical ones are due to Frobenius and Thompson. New results in this area have been obtained more recently by \textit{G.-X. Zhang} [Proc. Am. Math. Soc. 98, 579-582 (1986; Zbl 0607.20011)], \textit{A. Ballester-Bolinches} and \textit{X.-Y. Guo} [J. Algebra 228, No. 2, 491-496 (2000; Zbl 0961.20016)], and \textit{W. Bannuscher} and \textit{G. Tiedt} [Ann. Univ. Sci. Budapest Rolando Eötvös, Sect. Math. 37, 9-12 (2004; Zbl 0830.20038)]. In this paper the author gets improvements and extensions of some of the results of the above papers. His main achievement is the following theorem: Let \(P\) be a Sylow \(p\)-subgroup of a finite group \(G\). If \(p=2\), assume \(P\) is quaternion-free. Denote by \(D(G)\) the nilpotent residual of \(G\). Then the assertion that \(G\) is \(p\)-nilpotent is equivalent to: (a) \(N_G(P)\) is \(p\)-nilpotent and \(\Omega_1(D(G)\cap P\cap P^x)\leq Z(P)\) \(\forall x\in G\setminus N_G(P)\). (b) \(N_G(P)\) is \(p\)-nilpotent and \(|\Omega_1(D(G)\cap P\cap P^x)|\leq p^{p-1}\) \(\forall x\in G\setminus N_G(P)\). (c) \(\Omega_1(D(G)\cap P)\leq Z(N_G(P))\).