\(G\)-covering subgroup systems for the classes of supersoluble and nilpotent groups. (Q1425644)

Given a class of groups \(\mathfrak F\), a set \(\Sigma\) of subgroups of \(G\) is called an \(\mathfrak F\)-covering subgroup system of \(G\) if \(G\in{\mathfrak F}\) whenever every subgroup in \(\Sigma\) is in \(\mathfrak F\). The authors are concerned with non-trivial sets of subgroups of a finite group \(G\) which are \(\mathfrak F\)-covering subgroup systems for the classes \(\mathfrak F\) of all supersoluble and all nilpotent subgroups. In this paper, it is shown that given a soluble group \(G\) with a normal subgroup \(N\) such that \(G/N\) is supersoluble, a set \(\Sigma\) of subgroups of \(G\) is a covering subgroup system for the class of all supersoluble groups if for every maximal subgroup \(M\) of any Sylow subgroup of \(F(N)\), either \(M\) is normal in \(G\) or \(\Sigma\) contains a supplement of \(M\) in \(G\). A similar result is obtained when the condition is imposed on the maximal subgroups \(M\) of the Sylow subgroups of \(N\). From these results, some covering subgroup systems for the classes of all supersoluble and all nilpotent groups are derived. Finally, it is proved that if a set \(\Sigma\) of subgroups of a group \(G\) satisfies that for every cyclic subgroup \(L\) of \(G\) of prime order or order \(4\) not contained in the Frattini subgroup of any other subgroup of \(G\), either \(L\) is normal in \(G\) or \(\Sigma\) contains a supplement of \(L\) in \(G\), then \(\Sigma\) is a covering subgroup system for the class of supersoluble groups.