Normal \(\pi\)-complements for finite groups. (Q949851)

For a finite \(p\)-group \(P\) let \(\Omega_1(P)\) denote the subgroup generated by all elements of order \(p\). As it was proved by \textit{G. Zhang} [Proc. Am. Math. Soc. 98, 579-582 (1986; Zbl 0607.20011)], if \(P\) is a Sylow \(p\)-subgroup of a finite group \(G\) such that \(\Omega_1(P)\leq Z(P)\), and \(N_G(Z(P))\) has a normal \(p\)-complement, then \(G\) has a normal \(p\)-complement. In the paper under review this theorem of Zhang's is generalized as follows. Let \(P\) be a Sylow \(p\)-subgroup (or a nilpotent \(\pi\)-Hall subgroup) of a finite group \(G\), and assume that \(\Omega_1(P\cap P^x)\leq Z(P)\) for each \(x\in G\setminus N_G(P)\). If \(N_G(Z(P))\) has a normal \(p\)-complement (resp. \(\pi\)-complement), then \(G\) has a normal \(p\)-complement (resp. \(\pi\)-complement). Furthermore, if \(P\) has no section isomorphic to the quaternion group of order 8, and \(N_G(P)\) has a normal \(p\)-complement (resp. \(\pi\)-complement), then \(G\) has a normal \(p\)-complement (resp. \(\pi\)-complement).