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On the Fitting Height of Soluble Groups

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On the Fitting Height of Soluble Groups ON THE FITTING HEIGHT OF SOLUBLE GROUPS by GLEN STEVEN COLLINS A thesis submitted to The University of Birmingham for the degree of Doctor of Philosophy School of Mathematics The University of Birmingham 28th March 2014 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties.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. Abstract 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 treat- ments 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 pn-cycles and find 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 Zr be cyclic of prime order act on the extraspecial group F s1 2n such that F F; R , and suppose that RF acts on a group G such that+ CG F ≅ 1 and r; G 1. Then we show that F CG R F G . In≅ particular, when n r =s[ 1, then] f CG R f G . Fourthly, we show that there is no absolute( ) = bound( onS S) 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. Acknowledgements I have thoroughly enjoyed the past three and a half years and will look back on my time as a PhD student at Birmingham with great fondness. There are many people who have contributed to this enjoyment and to whom I would like to express my deep gratitude. My foremost thanks go to my supervisor Paul Flavell for his patient help and encouragement. His knowledge and insight have been invaluable in guiding me through my research. I also thank him for his own contributions to group theory and for helping me to understand some of the fascinating work that he has done. Further to Paul, I would like to thank all of the staff in the School of Mathematics. In particular, the members of the Algebra group, many of whom have given courses which I have attended during my time here, from which I have not only learnt a great deal but have taken much enjoyment. I thank Chris Parker for his continual interest in the problems that I am working on, and for taking time to listen to my ideas for solving them. I am also grateful to my examiners, Inna Capdeboscq and Simon Goodwin, for their feedback which has helped improve my work. I would also like to thank all of the postgraduates who have been here during my time at Birmingham. In particular, the members of office 315 who have made it a pleasant and enjoyable place to work. Special thanks go to Kim and to my parents, Steven and Dawn, for their encouragement and support. Finally, I gratefully acknowledge the financial support of the Engineering and Physical Sciences Research Council. i Contents Introduction 1 1 Background material 5 1.1 General group theoretic results . 5 1.2 Soluble, nilpotent and p-groups . 8 1.2.1 p-groups and π-groups . 8 1.2.2 Nilpotent groups . 11 1.2.3 Soluble groups . 13 1.2.4 Frattini subgroup . 14 1.3 Fitting height . 16 1.4 Generalised Fitting subgroup . 20 1.5 Coprime action . 22 1.6 Special groups and symplectic forms . 23 1.7 Permutation and Frobenius groups . 24 1.8 Representation theory . 26 1.9 Free modules . 30 1.10 First cohomology group . 33 2 Properties of 2-generated subgroups 35 2.1 Preliminary results . 38 2.2 Properties of 2-generated soluble subgroups . 40 2.3 Properties of 2-generated subgroups . 46 2.4 Concluding remarks . 64 3 Soluble primitive permutation groups 66 3.1 Preliminary results . 67 3.2 Basic structure . 72 3.3 Soluble primitive permutation groups generated by two pn-cycles . 75 3.4 Bounding the Fitting height in terms of n and p . 88 ii 3.5 A further application for this bound . 92 3.6 Concluding remarks . 93 4 On fixed-point-free automorphisms 94 4.1 Preliminary results . 96 4.2 Proof of the main theorem . 101 4.3 Further work . 115 4.4 The dihedral group as a group of automorphisms . 116 5 Groups with two Sylow numbers 119 5.1 Preliminaries . 119 5.2 Constructing examples . 121 6 On Hall normally embedded subgroups 129 6.1 Preliminaries . 130 6.2 Proofs of the main results . 132 6.3 Related work . 140 Bibliography 145 iii Introduction In mathematics, we often find series of inclusions of types of mathematical objects. For example, in module theory we have free modules projective modules flat modules modules; and in analysis we have⊂ ⊂ ⊂ Hilbert spaces Banach spaces metric spaces topological spaces: Let A be a mathematical⊂ object of type⊂ X and B a⊂ mathematical object of type Y . Suppose further that A is not of type Y but B, by virtue of being type Y , is automatically of type X. So mathematical objects of type Y mathematical objects of type X: A natural question to ask is: is there⊂ a well defined way of measuring how far A is from being a mathematical object of type Y ? So for example, let R be a ring and M an R-module which is not flat. Is there a well defined way of measuring how far M is from being a flat R-module? Indeed there is, and the answer lies in the concept of flat dimension. For any R-module, say X, there exist flat R-modules Xi and an exact sequence (possibly of infinite length) dn+1 dn d2 d1 d0 ::: Xn ::: X1 X0 X 0: Such a sequence is called a flat resolution of X. If there exists an n such ÐÐ→ Ð→ Ð→ Ð→ Ð→ → that Xn 1 0, then X has finite flat dimension, and the smallest such n is called the flat dimension of X. The flat dimension of a module can be seen + as a measure= of how far a module is from being flat. We can also define projective and free resolutions by replacing the Xi with projective and free R-modules respectively. Furthermore, for any R-module X, we can always form projective and free resolutions of X, and thus have well defined concepts of both projective and free dimension. These can be seen as measures of how far an R-module is from being projective or free respectively. 1 So what about in finite group theory? If we have an inclusion of groups groups of type Y groups of type X; then how do we measure how far any⊂ given group G of type X is from being a group of type Y . The most common approach is as follows: look for subgroups Gi such that 1 G0 G1 G2 ::: Gn G where Gi Gi 1 are groups= of type⊴ ⊴Y for⊴ 1 ⊴i n.= If we are guaranteed the existence of such G , then we could search for the shortest chain and take − i this to be~ a `measure' of how far G is being≤ a group≤ of type Y . Of course, this may not be possible. If we take an arbitrary finite group and wish to know how far it is from being abelian, then we might be at a loss by adopting this approach. Indeed, we would fail if we tried this with any nontrivial perfect group. However, there are many instances where this approach is successful. Consider the following inclusion nilpotent groups soluble groups. For any soluble group G, it is possible⊂ to find subgroups Gi such that 1 G0 G1 ::: Gn G where Gi Gi 1 is a nilpotent= group⊴ for⊴ all⊴ 1 =i n. The smallest such n such that G possesses a series as above is called the Fitting height of G. As − such, we can~ think of Fitting height as a measure≤ ≤ of how far a soluble group is from being nilpotent. Being able to break groups up in this way can often be very useful. It can allow us to break problems down about certain classes of groups to questions about other classes of groups, which may be easier to understand, and how these groups can be `put together'. This is quite a generic idea in finite group theory and has often been the motivation behind research projects. The Jordan-H¨oldertheorem is one of the most fundamental results in group theory and states that: given any finite group G and two series for G 1 H0 H1 ::: Hn G; 1 K0 K1 ::: Km G such that Hi=Hi 1⊴and⊴Kj K⊴ j 1 are= simple= for⊴ all⊴ 1 i⊴ n and= 1 j m, then n m and H H 1 i n K K 1 j m .
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