Abstract
I want to give here a rather biased account of recent work in the theory of classes of finite soluble groups. I will be concentrating on results which have something to say about the classes themselves, rather than results which use the classes to obtain a picture of the internal structure of finite soluble groups. My main excuse for doing so is that this part of the theory is at a very interesting stage: the classes are proving to be more exotic than might have been expected, and though we know little about them, some results and techniques are appearing, and it seems likely we will not remain so ignorant for long.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Reinhold Baer, “Classes of finite groups and their properties”, Illinois J. Math. 1 (1957), 115–187. MRI9,386.
Donald W. Barnes and Otto H. Kegel, “Geschütz functors on finite soluble groups”, Math. 2. 94 (1966), 134–142. MR34#4350.
J.G. Beidleman and A.R. Makan, “On normal Fischer classes”, bull. London Math. Soc. 5 (1973), 100–102.
Dieter Blessenohl und Wolfgang Gaschütz, “Cher normale Schunck-und Fittingklassen”, Math. Z. 118 (1970), 1–8. MR43#3344.
R.M. Bryant, R.A. Bryce and B. Hartley, “The formation generated by a finite group”, Bull. Austral. Math. Soc. 2 (1970), 347–357. MR43#4901.
R.A. Bryce and John Cossey, “Fitting formations of finite soluble groups”, Math. Z. 127 (1972), 217–223. Zb1.229.20011.
R.A. Bryce and John Cossey, “Metanilpotent Fitting classes”, J. Austral. Math. Soc. (to appear).
R.A. Bryce and John Cossey,’“Subdirect product closed Fitting classes”, these Proc.
R.A. Bryce and John Cossey, “Maximal Fitting classes of finite soluble groups Bull. Austral. Math. Soc. 10 (1974)
A.R. Camina, “A note on Fitting classes”, submitted.
R.W. Carter, “On a class of finite soluble groups”, Proc. London Math. Soc. (3) 9 (1959), 623–640. MR22#5677.
Roger W. Carter, “Nilpotent self-normalizing subgroups of soluble groups”, Math. Z. 75 (1961), 136–139. MR23#A928.
R.W. Carter, “Nilpotent self-normalizing subgroups and system normalizers”, Rroc. London Math. Soc. (3) 12 (1962), 535–563. MR25#3988.
Edward Cline, “On an embedding property of generalized Carter subgroups”, Pacific J. Math. 29 (1969), 491–519. MR42#7773.
John Cossey, “Products of Fitting classes”, submitted.
John Cossey and Sheila Oates Macdonald, “On the definition of saturated formations of groups”, Bull. Austral. Math. Soc. 4 (1971), 9–15. MR43#2084.
R.S. Dark, “On subnormal embedding theorems for groups”, J. London Math. Soc. 43 (1968), 387–390. MR37#1446.
Rex S. Dark, “Some examples in the theory of injectors of finite soluble groups”, Math. Z. 127 (1972), 145–156. Zb1.226.20013.
K. Doerk, “Ober nomomorphe auflösbarer Gruppen”, submitted.
K. Doerk, “Metanilpotente Fittingschunckklassen”, submitted.
B. Fischer, Habilitationsschrift (Universität Frankfurt).
Bernd Fischer, Classes of conjugate subgroups in finite solvable groups ( Lecture Notes, Yale University, 1966 ).
B. Fischer, W. Gaschütz und B. Hartley, “Injektoren endlicher auflösbarer Gruppen”, Math. Z. 102 (1967), 337–339. MR36#6504.
Wolfgang Gaschütz, “Zur Theorie der endlichen auflösbaren Gruppen”, Math. Z. 80 (1963), 300–305. MR31#3505.
W. Gaschütz, Selected topics in the theory of soluble groups (Lectures given at the 9th Summer Research Institute of the Australian Mathematical Society, Canberra, 1969. Notes by J. Looker).
W. Gaschütz, “Zwei Bemerkungen über normale Fittingklassen”, J. Algebra (to appear).
Wolfgang Gaschütz und Ursula Lubeseder, “Kennzeichnung gesättigter Formationen”, Math. Z. 82 (1963), 198–199. MR27#5817.
P. Hall, “A characteristic property of soluble groups”, J. London Math. Soc. 12 (1937), 198-200. FdM63,69.
P. Hall, “The splitting properties of relatively free groups”, Proc. London Math. Soc. (3) 4 (1954), 343–356. MR16,217.
B. Hartley, “On Fischer’s dualization of formation theory”, Proc. London Math. Soc. (3) 19 (1969), 193–207. MR39#5696.
Trevor O. Hawkes, “On Fitting formations”, Math. Z. 117 (1970), 177–182. MR4343346.
Trevor Hawkes, “Skeletal classes of soluble groups”, Arch. der Math. 22 (1971), 577–589. MR46#248.
T.O. Hawkes, “Closure operations for Schunck classes”, J. Austral. Math. Soc. 16 (1973), 316–318.
I.O. Hawkes, in preparation.
B. Ruppert, Endliche Gruppen S (Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967 ). MR37#302.
Hans Lausch, “On normal Fitting classes”, Math. Z. 130 (1973), 67–72. Z61.238.20039.
F. Peter Lockett, “On the theory of Fitting classes of finite soluble groups”, (PhD thesis, University of Warwick, Coventry, 1971 ).
F. Peter Lockett, “On the theory of Fitting classes of finite soluble groups”, Math. Z. 131 (1973), 103–115.
F.P. Lockett, “An example in the theory of Fitting classes”, Bull. London Math. Soc. 5 (1973), 271–274.
F. Peter Lockett, “The Fitting class F* ”, submitted.
Ursula Lubeseder, “Formationsbildungen in endlichen auflösbaren Gruppen”, (Dissertation, Kiel, 1963 ).
A.R. Makan, “On some aspects of finite soluble groups”, PhD thesis, Australian National University, Canberra, 1971. Abstract in Bull. Austral. Math. Soc. 6 (1972), 157–158.
A.R. Makan, “Fitting classes with the wreath product property are normal”, submitted.
Peter M. Neumann, “A note on formations of finite nilpotent groups”, Bull. London Math. Soc. 2 (1970), 91. MR41#5470.
Hermann Schunck, “H-Untergruppen in endlichen auflösbaren Gruppen”, Math. Z. 97 (1967), 326–330. MR35#254.
James Wiegold, “Schunk classes are nilpotent product closed”, Bull. Austral. Math. Soc. 1 (1969), 27–28. MR40#1478.
Graham John Wood, “On generalizations of Sylow theory in finite soluble groups”, (PhD thesis, Queen Mary College, University of London, 1973 ).
Graham J. Wood, “A lattice of homoraorphs”, Math. Z. 130 (1973), 31–37. Zb1.239.20016.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cossey, J. (1974). Classes of Finite Soluble Groups. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-662-21571-5_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06845-7
Online ISBN: 978-3-662-21571-5
eBook Packages: Springer Book Archive