Frobenius groups of automorphisms and their fixed points

Khukhro, Evgeny and Makarenko, Natalia and Shumyatsky, Pavel (2014) Frobenius groups of automorphisms and their fixed points. Forum Mathematicum, 26 (1). pp. 73-112. ISSN 0933-7741

Full content URL: http://dx.doi.org/10.1515/form.2011.152

Documents
Frobenius groups of automorphisms and their fixed points
[img]
[Download]
[img]
Preview
PDF
__ddat02_staffhome_jpartridge_14forum-khu-mak-shu.pdf - Whole Document

393kB
Item Type:Article
Item Status:Live Archive

Abstract

Suppose that a finite group G admits a Frobenius group of automorphisms with kernel F and complement H such that the fixed-point subgroup of F is trivial: . In this situation various properties of G are shown to be close to the corresponding properties of . By using Clifford's theorem it is proved that the order is bounded in terms of and , the rank of G is bounded in terms of and the rank of , and that G is nilpotent if is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of G in the case of metacyclic . The exponent of G is bounded in terms of and the exponent of by using Lazard's Lie algebra associated with the Jennings–Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of G is bounded in terms of and the nilpotency class of by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.

Keywords:Frobenius group, Automorphism, Finite group, Exponent, Lie ring, Lie algebras, Lie group, Graded, Solvable, Nilpotent, NotOAChecked
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:16202
Deposited On:05 Dec 2014 11:01

Repository Staff Only: item control page