Frobenius groups of automorphisms and their fixed points

Khukhro, Evgeny, 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

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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

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