Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T02:59:31.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds on the fitting length of finite soluble groups with supersoluble Sylow normalisers

Published online by Cambridge University Press:  17 April 2009

R.A. Bryce ,
V. Fedri  and
L. Serena
R.A. Bryce
Affiliation:
Department of Mathematics, Faculty of Science, Australian National University, GPO Box 4, Canberra City ACT 2601, Australia
V. Fedri
Affiliation:
Department of Mathematics, Faculty of Science, Australian National University, GPO Box 4, Canberra City ACT 2601, Australia
L. Serena
Affiliation:
Istituto Matematico, Università degli Studi di Firenze, via Morgagni, 67/A 50135 Firenze, Italy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that, in a finite soluble group, all of whose Sylow normalisers are super-soluble, the Fitting length is at most 2m + 2, where pm is the highest power of the smallest prime p dividing |G/Gs| here Gs is the supersoluble residual of G. The bound 2m + 2 is best possible. However under certain structural constraints on G/GS, typical of the small examples one makes by way of experimentation, the bound is sharply reduced. More precisely let p be the smallest, and r the largest, prime dividing the order of a group G in the class under consideration. If a Sylow p–subgroup of G/GS acts faithfully on every r-chief factor of G/GS, then G has Fitting length at most 3.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 1 , August 1991 , pp. 19 - 31
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Bianchi, M.G., Berta Mauri, A. Gillio and Hauck, P., ‘On finite groups with nilpotent Sylow normalizers’, Arch. Math. 47 (1986), 193–197.CrossRefGoogle Scholar
[2]Curtis, C.W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York and London, 1962).Google Scholar
[3]Fedri, V. and Serena, L., ‘Finite soluble groups with supersoluble Sylow normalizers’, Arch. Math. 50 (1988), 11–18.CrossRefGoogle Scholar
[4]Higman, G., ‘Complementation of abelian normal subgroups’, Publ. Math. Debrecen 4 (1955), 455–458.CrossRefGoogle Scholar
[5]Huppert, B. and Blackburn, N., Finite groups III: Grundlehren Math. Wiss. 242 (Springer-Verlag, Berlin, Heidelberg, New York, 1982).Google Scholar
[6]Kurzweil, H., ‘p–Automorphismen von auflösbaren p'-Gruppen’, Math. Z. 120 (1971), 326–354.CrossRefGoogle Scholar