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Fixed-point free action of an abelian group of odd non-squarefree exponent

Part of: Representation theory of groups

Published online by Cambridge University Press:  19 January 2011

Gülin Ercan ,
İsmail Ş. Güloğlu  and
Öznur Mut Sağdiçoğlu
Gülin Ercan
Affiliation:
Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey (ercan@metu.edu.tr; oznur@uekae.tubitak.gov.tr)
İsmail Ş. Güloğlu
Affiliation:
Department of Mathematics, Doğuş University, Istanbul, Turkey (iguloglu@dogus.edu.tr)
Öznur Mut Sağdiçoğlu
Affiliation:
Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey (ercan@metu.edu.tr; oznur@uekae.tubitak.gov.tr)
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Abstract

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Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding conjecture is that if (|G|, |A|) = 1, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. It is expected that the conjecture is true when the coprimeness condition is replaced by the assumption that A is nilpotent. We establish the conjecture without the coprimeness condition in the case where A is an abelian group whose order is a product of three odd primes and where the Sylow 2-subgroups of G are abelian.

Keywords

automorphisms of solvable groupsnon-coprime actionfixed-point free actionCarter subgroup

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D15: Nilpotent groups, $p$-groups 20D30: Series and lattices of subgroups 20D45: Automorphisms
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 77 - 89
Copyright
Copyright © Edinburgh Mathematical Society 2011

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