Groups and Lie rings with a Frobenius group of automorphisms

Iusa, Valentina (2019) Groups and Lie rings with a Frobenius group of automorphisms. PhD thesis, University of Lincoln.

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Groups and Lie rings with a Frobenius group of automorphisms
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Item Type:Thesis (PhD)
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Abstract

Suppose that a Frobenius group F H, with kernel F and complement H, acts by automorphisms on a finite group G. Work of E. Khukhro, N. Makarenko and P. Shumyatsky showed that many properties of G are influenced by the corresponding properties of the centraliser of H in G, and possibly by the order of H as well. In particular, they proved that if F is cyclic acting fixed-point-freely and the centraliser of H is nilpotent of class c, then the nilpotency class of G can be bounded from above by a function of c and the order of H.

In this thesis we prove that the dependence of the bound on the order of the Frobenius complement is essential, by constructing an explicit example. We also discuss a generalisation of the quoted result of Khukhro, Makarenko and Shumyatsky to the case of abelian non-cyclic kernels.

These results are obtained by application of different Lie ring methods. In the former case, the desired family of groups is constructed by using the Lazard correspondence. The latter result relies on properties of Lie rings admitting a grading over an abelian group with few non-trivial components and many commuting components.

The final chapter can be read independently of the rest of the thesis. In fact it presents our contribution to a different and independent problem, that is the classification of certain Lie algebras of maximal class

Divisions:College of Science > School of Mathematics and Physics
ID Code:47482
Deposited On:06 Dec 2021 10:45

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