Collins, Glen Steven (2014)
Ph.D. thesis, University of Birmingham.
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 AbstractWe consider five separate problems in finite group theory which cover a range of topics including properties of 2generated subgroups, permutation groups, fixedpointfree automorphisms and the study of Sylow structure. The treatments of these problems are largely selfcontained, 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 2generated, 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 fixedpointfree 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 HNEgroups as precisely those groups which split over their nilpotent residual, which itself is cyclic of squarefree order.

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