Khukhro, Evgeny, Makarenko, Natalia and Shumyatsky, Pavel
(2014)
Frobenius groups of automorphisms and their fixed points.
Forum Mathematicum, 26
(1).
pp. 73112.
ISSN 09337741
Full content URL: http://dx.doi.org/10.1515/form.2011.152
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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 fixedpoint 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 torsionfree locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for nonmetacyclic 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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