\(p\)-supersolvability of factorized finite groups. (English) Zbl 0804.20015

Carocca, Angel

Hokkaido Math. J. 21, No. 3, 395-403 (1992).

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.

Reviewer: H.Heineken (Würzburg)

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

Keywords:

mutually permutable subgroups; normal subgroup; \(p\)-supersoluble; \(p\)- nilpotent