Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T15:09:33.977Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected]; [email protected])
İsmail Ş. Güloğlu
Affiliation:
Department of Mathematics, Doğuş University, Istanbul, Turkey ([email protected])
Öznur Mut Sağdiçoğlu
Affiliation:
Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey ([email protected]; [email protected])
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.

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

References

1.Bell, S. D. and Hartley, B., A note on fixed-point-free actions of finite groups, Q. J. Math. 41(162) (1990), 127–130.CrossRefGoogle Scholar
2.Cheng, K. N., Finite groups admitting fixed point free automorphisms of order pq, Groups—St Andrews 1985, Proc. Edinb. Math. Soc. 30(1) (1987), 51–56.Google Scholar
3.Dade, E. C., Carter subgroups and Fitting heights of finite solvable groups, Illinois J. Math. 13 (1969), 449–514.CrossRefGoogle Scholar
4.Ercan, G. and Güloğlu, İ.Ş., Finite groups admitting fixed-point free automorphisms of order pqr, J. Group Theory 7(4) (2004), 437–446.CrossRefGoogle Scholar
5.Ercan, G. and Güloğlu, İ.Ş., Fixed point free action on groups of odd order, J. Alg. 320 (2008), 426–436.CrossRefGoogle Scholar
6.Isaacs, I. M., Fixed points and characters in groups with noncoprime operator groups, Can. J. Math. 20 (1968), 1315–1320.Google Scholar
7.Isaacs, I. M., Character theory of finite groups (Dover Publications, New York, 1994).Google Scholar
8.Shult, E., On groups admitting fixed point free abelian operator groups, Illinois J. Math. 9 (1965), 1315–1320.CrossRefGoogle Scholar
9.Turull, A., Fitting height of groups and of fixed points, J. Alg. 86 (1984), 555–566.CrossRefGoogle Scholar
10.Turull, A., Character theory and length problems: finite and locally finite groups, NATO Science Series C: Mathematical and Physical Sciences, Volume 471, pp. 377–400 (Kluwer, Dordrecht, 1995).Google Scholar