A question of R. Maier concerning formations
The paper answers affirmatively the following question of \textit{R. Maier}: ``Let \(\mathfrak f\) be a saturated formation. If \(G=HK\) with \(\mathfrak f\)-groups \(H\) and \(K\) such that every subgroup of \(H\) is permutable with every subgroup of \(K\), then \(G\) belongs to \(\mathfrak f\) provided that \(\mathfrak f\) contains the saturated formation of all supersoluble groups. Does the above result extend to non-saturated formations which contain all supersoluble groups? The main result of this paper is: Let \(\mathfrak f\) be a formation containing the class of all supersoluble groups. Suppose that the group \(G=HK\) is the product of the totally permutable subgroups \(H\) and \(K\) (i.e. every subgroup of \(H\) is permutable with every subgroup of \(K\)). If \(H\) and \(K\) belong to \(\mathfrak f\), then \(G\) belongs to \(\mathfrak f\).
scientific article; zbMATH DE number 922330