On the membership problem of finite groups in solvably saturated formations. (Q1929791)

All groups considered in this paper are finite. A formation \(\mathcal F\) is a solvably saturated formation if every group \(G\) having a solvable normal subgroup \(N\) for which \(G/\Phi(N)\in\mathcal F\) belongs to \(\mathcal F\). The main result in the paper is the following: Let \(\mathcal H\) be a solvably saturated subformation of a solvably saturated formation \(\mathcal F\). Let \(N\) be a solvable normal subgroup of a group \(G\), where \(G/N\in\mathcal F\). If every \(G\)-chief factor of \(F(G)/\Phi(G)\) is \(\mathcal H\)-central in \(G\), then \(G\in\mathcal F\).