On finite groups whose principal factors are simple groups. (English. Russian original) Zbl 0934.20016

Vasil’ev, A. F.; Vasil’eva, T. I.

Russ. Math. 41, No. 11, 8-12 (1997); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1997, No. 11, 10-14 (1997).

V. A. Vedernikov introduced in [Dokl. Akad. Nauk BSSR 32, No. 10, 872–875 (1988; Zbl 0663.20016)] a number of classes of compound groups, among these the class \({\mathfrak U}_c\) of \(c\)-supersoluble groups (i.e., groups which possess a principal series with all its factors being simple groups) and gave some properties of the class \({\mathfrak U}_c\), among them that \({\mathfrak U}_c\) forms an \(S_n\)-closed formation. This paper studies other properties of the formation of all \(c\)-supersoluble groups: the formation \({\mathfrak U}_c\) is not saturated, but for \({\mathfrak U}_c\) a close to saturation property can be derived from a result of the paper; the formation \({\mathfrak U}_c\) is not radical, but the authors obtain for \(c\)-supersoluble groups analogous results to the following known results: the group \(G=HK\), where \(H\) and \(K\) are normal supersoluble subgroups, is supersoluble if \((|G:|,|G:K|)=1\), or if \(G\) has nilpotent commutant.

Reviewer: R.Covaci (Cluj-Napoca)

MSC:

20D10	Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D30	Series and lattices of subgroups
20B40	Computational methods (permutation groups) (MSC2010)

Keywords:

supersoluble groups; principal series; formations; chief series

Citations:

Zbl 0663.20016