Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T08:34:52.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgroup closed Fitting classes are formations

Published online by Cambridge University Press:  24 October 2008

R. A. Bryce  and
John Cossey
R. A. Bryce
Affiliation:
The Australian National University, Canberra
John Cossey
Affiliation:
The Australian National University, Canberra
Get access Rights & Permissions [Opens in a new window]

Extract

Since their introduction by Fischer(12) and Fischer, Gaschütz and Hartley (13) Fitting classes of soluble groups have attracted attention on two fronts (all groups considered in this paper will be finite and soluble). On the one hand is their important role in the structure of finite soluble groups, a good account of which can be found in Gaschütz (14), and on the other is their intrinsic interest as classes of groups. This paper falls into the second category, and is a continuation and completion of (8). There we proved that a subgroup closed Fitting class is a formation if it consists of groups of nilpotent length at most three. Happily, at last, we can remove this qualification.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 2 , March 1982 , pp. 225 - 258
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bryant, Roger M.On S-critical groups. Quart. J. Math. Oxford (2), 22 (1971), 91–101.CrossRefGoogle Scholar
(2)Bryant, Roger M.Finite splitting groups in varieties of groups. Quart. J. Math. Oxford (2), 22 (1971), 169–172.CrossRefGoogle Scholar
(3)Bryant, Roger M., Bryce, R. A. and Habtley, B.The formation generated by a finite group. Bull Austral. Math. Soc. 2 (1970), 347–357.CrossRefGoogle Scholar
(4)Bryant, R. M. and Kovács, L. G.Tensor products of representations of finite groups. Bull. London Math. Soc. 4 (1972), 133–135.CrossRefGoogle Scholar
(5)Bryce, R. A.Protective groups in varieties. Bull. Austral. Math. Soc. 6 (1972), 169–175.CrossRefGoogle Scholar
(6)Bryce, R. A. and Cossey, John.Fitting formations of finite soluble groups. Math. Z. 127 (1972), 217–223.CrossRefGoogle Scholar
(7)Bryce, R. A. and Cossey, John.Metanilpotent Fitting classes. J. Austral. Math. Soc. 17 (1974), 285–304.CrossRefGoogle Scholar
(8)Bryce, R. A. and Cossey, John.Subgroup closed Fitting classes. Math. Proc. Cambridge Phil. Soc. 83 (1978), 195–204.CrossRefGoogle Scholar
(9)Carter, R., Fischer, B. and Hawkes, T.Extreme classes of finite soluble groups. J. Algebra 9 (1968), 285–313.CrossRefGoogle Scholar
(10)Curtis, Charles W. and Reiner, Irving.Representation theory of finite groups and associative algebras (pure and applied mathematics, No. 11 Interscience Publishers, New York and London, 1962).Google Scholar
(11)Dade, E. C.Characters of groups with normal extra-special subgroups. Math. Z. 152 (1976), 1–31.CrossRefGoogle Scholar
(12)Fischer, B.Klassen konjugierter Untergruppen in endlichen auflösbaren Gruppen (Habilitationsschrift Univ. Frankfurt (M), 1966).Google Scholar
(13)Fischer, B., Gaschütz, W. and Hartley, B.Injektoren endlicher auflösbarer Gruppen. Math. Z. 102 (1967), 337–339.CrossRefGoogle Scholar
(14)Gaschütz, W.Lectures on subgroups of Sylow type in finite soluble groups. Notes in Pure Mathematics 11 (Australian National University, Canberra, 1979).Google Scholar
(15)Hawkes, Trevor O.On Fitting formations. Math. Z. 117 (1970), 177–182.CrossRefGoogle Scholar
(16)Higman, G.Complementation of abelian normal subgroups. Publ. Math. Debrecen 4 (1955–1956), 455–458.CrossRefGoogle Scholar
(17)Huppert, B. Endliche Gruppen I. Die Grundlehren der mathematischen Wissenschaften, Bd 134 (Springer-Verlag, Berlin, Heidelberg, New York, 1967).Google Scholar
(18)Isaacs, I. M.Characters of solvable and symplectic groups. Amer. J. Math. 95 (1973), 594–635.CrossRefGoogle Scholar
(19)Isaacs, I. Martin.Character theory of finite groups (pure and applied mathematics No. 69, Academic Press, New York, San Francisco and London, 1976).Google Scholar
(20)Kochendörffer, R.Über treue irreduzible Darstellungen endlicher Gruppen. Math. Nachr. 1 (1948), 25–39.CrossRefGoogle Scholar
(21)Neumann, B. H.Twisted wreath products of groups. Arch. Math. 14 (1963), 1–6.CrossRefGoogle Scholar
(22)Neumann, Hanna. Varieties of Groups. Ergebnisse der Mathematik und ihrer Grenzgebeit (Springer-Verlag, Berlin, Heidelberg, New York, 1967).Google Scholar
(23)Neumann, Peter M.A note on formations of finite nilpotent groups. Bull. London Math. Soc. 2 (1970), 91.CrossRefGoogle Scholar
(24)Puttaswamaiah, B. M. and Dixon, John D.Modular representations of finite groups (Pure and Applied Mathematics No. 73, Academic Press, New York and London, 1977).Google Scholar