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On the fitting height of soluble groups

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.