\(p\)-saturated formations with complemented \(p\)-saturated subformations (Q1390471)

A formation of groups \(\mathcal F\) is said to be \(p\)-saturated if from \(G/L\in{\mathcal F}\), where \(L\subseteq O_p(G)\cap F(G)\), it always follows \(G\in{\mathcal F}\). There were found various characterizations of the classes of \(p\)-saturated formations. The paper describes \(p\)-saturated formations whose lattice of \(p\)-saturated subformations is Boolean. A subformation \(\mathcal M\) of a formation \(\mathcal F\) is said to be complemented in \(\mathcal F\) if \(\mathcal M\) is complemented in the lattice of subformations of the formation \(\mathcal F\). The main result of the paper is: Let \(\mathcal F\) be a \(p\)-saturated formation. Then the following conditions are equivalent: 1) \(\mathcal F\) is the direct product of its minimal \(p\)-saturated subformations; 2) the lattice of \(p\)-saturated subformations of the formation \(\mathcal F\) is Boolean; 3) in \(\mathcal F\) all minimal \(p\)-saturated subformations are complemented.