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Projektive Klassen endlicher Gruppen. I: Schunck- und Gaschützklassen. (English) Zbl 0544.20015
The author ingeniously extends the theory of W. Gaschütz [Selected topics in the theory of soluble groups. Lectures given at the 9th Summer Research Institute of the Australian Math. Soc. (Canberra 1969)] and H. Schunck [Math. Z. 97, 326-330 (1967; Zbl 0158.02802)] on special classes of finite soluble groups (formations and Schunck classes) and their corresponding canonical conjugacy classes of \({\mathcal H}\)-covering subgroups and \({\mathcal H}\)-projectors in finite soluble groups to the universe of arbitrary finite groups. A general theory of projectors and covering subgroups is developed and existence and conjugacy criteria are given for such subgroups. The main point of the paper is to introduce a special sort of \({\mathcal H}\)-subgroups “between \({\mathcal H}\)-projectors and \({\mathcal H}\)-covering subgroups” which, like \({\mathcal H}\)-projectors, always exist for Schunck classes, but, like \({\mathcal H}\)-covering subgroups, have heredity properties ensuring conjugacy criteria. Finally, the paper successfully deals with the question when the projectors have always the above heredity properties.
Reviewer: R.Covaci

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D25 Special subgroups (Frattini, Fitting, etc.)
20D30 Series and lattices of subgroups
Full Text: DOI EuDML
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