On \(G\)-covering subgroup systems of finite groups. (Q663089)

Let \(\mathcal F\) be a class of groups and \(G\) be a finite group. A set \(\Sigma\) of subgroups of \(G\) is called a \(G\)-covering subgroup system for the class \(\mathcal F\) if \(\Sigma\subseteq\mathcal F\) implies \(G\in\mathcal F\). Let \(p\) be an odd prime dividing the size of a finite group \(G\). For a \(p\)-Sylow subgroup \(P\) of \(G\), Thompson's \(p\)-nilpotence criterion states that the set \(\{N_G(J(P)),C_G(Z(G))\}\) is a \(G\)-covering subgroup system for the class of all \(p\)-nilpotent groups. By the Glauberman-Thompson theorem the set \(\{N_G(Z(J(P)))\}\) is a \(G\)-covering subset for the class of all \(p\)-nilpotent groups. Let \(\Sigma\) be the set of all normalizers of all Sylow subgroups of \(G\). Then \(\Sigma\) is a \(G\)-covering subgroup system for the class of all nilpotent groups by \textit{M.~Bianchi, A.~Gillio Berta Mauri} and \textit{P.~Hauck} [Arch. Math. 47, 193-197 (1986; Zbl 0605.20017)], but not for the class of all supersoluble groups (see \textit{V.~Fedri} and \textit{L.~Serena} [Arch. Math. 50, No. 1, 11-18 (1988; Zbl 0638.20013)]). The paper continues this approach by considering the following sets of subgroups. Let \(p,q\) be distinct primes dividing the size of a finite group \(G\) and let \(P\) be a \(p\)-Sylow subgroup of \(G\). A set of subgroups of \(G\) containing at least one supplement in \(G\) to each maximal subgroup of \(P\) will be denoted by \(\Sigma_p\). The notation \(\Sigma_q\) is defined similarly. The main result is the following. Theorem A. The set \(\Sigma_p\cup\Sigma_q\) is a \(G\)-covering subgroup system for the classes of all metanilpotent, all supersoluble, all \(p\)-supersoluble, all \(p\)-nilpotent and all Abelian groups.