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On the supersolvability of finite groups. (English) Zbl 0685.20018
The object of this paper is to find sufficient conditions for the finite group \(G=HK\), the product of two subgroups, to be supersolvable. The main sets of conditions are: (1) \(H\) and \(K\) are supersolvable and each subgroup of \(H\) is quasinormal in \(K\) (\(H\) is quasinormal in \(K\) if \(HL=LH\) for all subgroups \(L\) of \(K\)); (2) \(H\) is nilpotent, \(K\) is supersolvable and each is quasinormal in the other; (3) \(H\) and \(K\) are supersolvable, have coprime indices, for each pair of primes \(p,q\) with \(p>q\), \(p\mid|G:H|\), \(q\mid|G:K|\), then \(p\not\equiv 1(q)\), and each is quasinormal in the other; (4) \(G'\) is nilpotent and each of \(H,K\) is supersolvable and quasinormal in the other. These results generalize work of R. Baer [Ill. J. Math. 1, 115-187 (1957; Zbl 0077.03003)], D. K. Friesen [Proc. Am. Math. Soc. 30, 46-48 (1971; Zbl 0232.20037)] and O. H. Kegel [Math. Z. 87, 42-48 (1965; Zbl 0123.02503)].
Reviewer: J.D.P.Meldrum

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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