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On the fitting height of soluble groups

Collins, Glen Steven (2014)
Ph.D. thesis, University of Birmingham.

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We consider five separate problems in finite group theory which cover a range of topics including properties of 2-generated subgroups, permutation groups, fixed-point-free automorphisms and the study of Sylow structure. The treatments of these problems are largely self-contained, but they all share an underlying theme which is to study finite soluble groups in terms of their Fitting height.
Firstly, we prove that if A is a maximal subgroup of a group G subject to being 2-generated, and V <\(_-\) G is a nilpotent subgroup normalised by A, then F*(A)V is quasinilpotent. Secondly, we investigate the structure of soluble primitive permutation groups generated by two p\(^n\)-cycles and upper bounds for their Fitting height in terms of p and n. Thirdly, we extend a recent result regarding fixed-point-free automorphisms. Namely, let R \(\thicksim\)\(_=\) Z\(_r\) be cyclic of prime order act on the extraspecial group F \(\thicksim\)\(_=\) s\(^1\)\(^+\)\(^2\)\(^n\) such that F = [F,R], and suppose that RF acts on a group G such that C\(_G\)(F) = 1 and (r, |G| = 1. Then we show that F(C\(_G\)R)) \(\subseteq\) F(G). In particular, when r x sn+1, then f(C\(_G\)(R)) = f(G). Fourthly, we show that there is no absolute bound on the Fitting height of a group with two Sylow numbers. Lastly, we characterise partial HNE-groups as precisely those groups which split over their nilpotent residual, which itself is cyclic of square-free order.

Type of Work:Ph.D. thesis.
Supervisor(s):Flavell, Paul
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:Department of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:5244
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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