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 |
References:
| [1] | R. Baer, Classes of finite groups and their properties. Illinois J. Math.1, 115-187 (1957). · Zbl 0077.03003 |
| [2] | D. R. Friesen, Products of normal supersolvable subgroups. Proc. Amer. Math. Soc.30, 46-48 (1971). · Zbl 0232.20037 · doi:10.1090/S0002-9939-1971-0280590-4 |
| [3] | O. H. Kegel, Zur Struktur mehrfach faktorisierbarer endlicher Gruppen. Math. Z.87, 42-48 (1965). · Zbl 0123.02503 · doi:10.1007/BF01109929 |
| [4] | D.Gorenstein, Finite groups. New York 1968. · Zbl 0185.05701 |
| [5] | K. Doerk, Minimal nicht überauflösbare, endliche Gruppen. Math. Z.91, 198-205 (1966). · Zbl 0135.05401 · doi:10.1007/BF01312426 |
| [6] | M.Hall, The theory of groups. New York 1959. · Zbl 0084.02202 |
| [7] | B.Huppert, Endliche Gruppen I. Berlin-Heidelberg-New York 1967. · Zbl 0217.07201 |
| [8] | W. R.Scott, Group theory. Englewood Cliffs, New Jersey 1964. |
| [9] | B. Huppert, Monomiale Darstellung endlicher Gruppen. Nagoya Math. J.6, 93-94 (1953). · Zbl 0053.01201 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
