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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
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Abstract

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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

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