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
PhD Thesis
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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.

Keywords:Lie algebra, Lazard correspondence
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:48497
Deposited On:09 Mar 2022 10:00

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