Problems on product of formations (Q697403)

All groups considered are finite. A formation \(\mathfrak F\) is said to be \(p\)-saturated if it contains every group \(G\) such that \(G/O_p(G)\cap\Phi(G)\in\mathfrak F\), where \(O_p(G)\) is the maximal normal \(p\)-subgroup of \(G\) and \(\Phi(G)\) is the Frattini subgroup of \(G\). A formation \(\mathfrak F\) is called \(w\)-saturated if \(\mathfrak F\) is \(p\)-saturated for every \(p\in w\). A formation \(\mathfrak F\) is called a one-generated \(w\)-saturated formation if \(\mathfrak F\) is the intersection of all \(w\)-saturated formations containing some group \(G\in\mathfrak F\). The product \(\mathfrak{MH}\) of formations \(\mathfrak M\) and \(\mathfrak H\) is the class \((G\mid G^{\mathfrak H}\in\mathfrak M)\). In this paper it is proved the following Theorem. Let the product \(\mathfrak F=\mathfrak{MH}\) of two formations \(\mathfrak M\) and \(\mathfrak H\) be a one-generated \(w\)-saturated formation. If \(\mathfrak F\neq\mathfrak H\), then \(\mathfrak M\) is also a \(w\)-saturated formation. This theorem is an extension of the main result of the first author's previous paper [Commun. Algebra 28, No.~10, 4767-4782 (2000; Zbl 0969.20013)]. By using this result, it is given an answer to a problem proposed by \textit{A. N. Skiba} and \textit{L. A. Shemetkov} [see On partially local formations. Dokl. Akad. Nauk Belarusi 39, No. 3, 9-11 (1995)].