Some results on \(p\)-nilpotence and solubility of finite groups
Let \(G\) be a finite group, \(P\) a Sylow \(p\)-subgroup of \(G\), and let \(A\) be a \(p\)-group. The subgroup of \(A\) generated by its elements of order \(p^i\) is denoted by \(\Omega_i(A)\). For \(p=2\), \(\Omega(A)\) stands for the subgroup of \(A\) generated by \(\Omega_1(A)\) and \(\Omega_2(A)\), while otherwise \(\Omega(A)=\Omega_1(A)\). If \(\Omega(P\cap G')\subseteq Z(N_G(P))\), then \(G\) is \(p\)-nilpotent (Theorem 1). The toughest proof of the paper is that of Theorem 2: If \(p=2\), \(P\) is quaternion-free, \(N_G(P)\) is 2-nilpotent, and \(\Omega_1(P\cap G')\subseteq Z(P)\), then \(G\) is 2-nilpotent. Finally, if \(M\) is a nilpotent maximal subgroup of \(G\), \(Q\) a Sylow 2-subgroup of \(M\) and \(\Omega(Q\cap G')\subseteq Z(Q)\), then \(G\) is soluble (Theorem 3).
scientific article; zbMATH DE number 1471824
scientific article