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\(p\)-supersolvability of factorized finite groups. (English) Zbl 0804.20015
The author calls two subgroups \(H\), \(K\) of a group mutually permutable if \(H\) is permutable with every subgroup of \(K\) and \(K\) is permutable with every subgroup of \(H\). He obtains the following main results: If \(G = HK \neq 1\) and \(H\) and \(K\) are mutually permutable, then \(H\) or \(K\) contains a nontrivial normal subgroup of \(G\) or \(F(G) \neq 1\) (Theorem A). If \(G = HK\) and \(H\) and \(K\) are \(p\)-supersoluble and mutually permutable, if further \(G'\) is \(p\)-nilpotent, then \(G\) is \(p\)-supersoluble (Theorem B). – If \(G = HK\), where \(H\) and \(K\) are mutually permutable, \(H\) is \(p\)- supersoluble and \(K\) is \(p\)-nilpotent, then \(G\) is \(p\)-supersoluble.

MSC:
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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