×

zbMATH — the first resource for mathematics

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

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Baer, R.: Classes of finite groups and their properties. Illinois J. Math.1, 115-187 (1957) · Zbl 0077.03003
[2] Baer, R., Förster, P.: Einbettungsrelationen und Formationen endlicher Gruppen. Erscheint
[3] Bender, H.: On groups with abelian Sylow 2-subgroups. Math. Z.117, 164-176 (1970) · Zbl 0225.20012
[4] Blessenohl, D., Laue, H.: Fittingklassen endlicher Gruppen, in denen gewisse Hauptfaktoren einfach sind. J. Algebra56, 516-532 (1979) · Zbl 0416.20015
[5] Carter, R.W., Fischer, B., Hawkes, T.O.: Extreme classes of finite soluble groups. J. Algebra9, 285-313 (1968) · Zbl 0177.03902
[6] Doerk, K.: Über Homomorphe endlicher auflösbarer Gruppen. J. Algebra30, 12-30 (1974) · Zbl 0346.20012
[7] Erickson, R.P.: Projectors of finite groups. Comm. Algebra10, 1919-1938 (1982) · Zbl 0498.20018
[8] Förster, P.: Über Projektoren und Injektoren in endlichen auflösbaren Gruppen. J. Algebra49, 606-620 (1977) · Zbl 0371.20021
[9] Gagen, T.M.: Topics in Finite Groups. Cambridge: Cambridge University Press 1976 · Zbl 0324.20013
[10] Gaschütz, W.: Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden. Math. Z.60, 274-286 (1954) · Zbl 0056.02401
[11] Gaschütz, W.: Selected topics in the theory of soluble groups. Lectures given at the 9th Summer Research Institute of the Australian Mathematical Society (Canberra 1969)
[12] Hawkes, T.O.: On formation subgroups of a finite soluble group. J. London Math. Soc.44, 243-250 (1969) · Zbl 0174.31001
[13] Hawkes, T.O.: Closure operations for Schunck classes. J. Austral. Math. Soc.16, 316-318 (1973) · Zbl 0298.20014
[14] Hawkes, T.O.: Two applications of twisted wreath products to finite soluble groups. Trans. Amer. Math. Soc.214, 325-335 (1975) · Zbl 0345.20022
[15] Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967 · Zbl 0217.07201
[16] Huppert, B., Blackburn, N.: Finite Groups II. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0477.20001
[17] Lafuente, J.: Clases de Schunck normales y clases derivadas. Dissertation. Departamento de Algebra y Fundamentos, Facultad de Ciencias, Universidad de Zaragoza 1977
[18] Lafuente, J.: Nonabelian crowns and Schunck classes of finite groups. Erscheint im Arch. Math. (Basel) · Zbl 0509.20011
[19] Schunck, H.:H-Untergruppen in endlichen auflösbaren Gruppen. Math. Z.97, 326-330 (1967) · Zbl 0158.02802
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.