ON CERTAIN GROUPS WITH FINITE CENTRALISER DIMENSION A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2020 Ulla Karhum¨aki Department of Mathematics Contents Abstract5 Declaration7 Copyright Statement8 Acknowledgements9 1 Introduction 14 1.1 Structure of this thesis........................... 19 2 Group-theoretic background material 23 2.1 Notation and elementary group theory.................. 23 2.2 Groups with finite centraliser dimension................. 28 2.3 Linear algebraic groups........................... 30 2.3.1 Groups of Lie type......................... 32 2.3.2 Automorphisms of Chevalley groups................ 34 2.4 Locally finite groups............................ 35 2.4.1 Frattini Argument for locally finite groups of finite centraliser dimension.............................. 36 2.4.2 Derived lengths of solvable subgroups of locally finite groups of finite centraliser dimension..................... 37 2.4.3 Simple locally finite groups..................... 37 3 Some model theory 41 3.1 Languages, structures and theories.................... 41 3.2 Definable sets and interpretability..................... 44 2 3.2.1 The space of types......................... 48 3.3 Stable structures.............................. 48 3.3.1 Stable groups............................ 50 3.4 Ultraproducts and pseudofinite structures................ 52 4 Locally finite groups of finite centraliser dimension 55 4.1 The structural theorem........................... 55 4.1.1 Control of sections......................... 56 4.1.2 Quasisimple locally finite groups of Lie type........... 58 4.1.3 Proof of Theorem 4.1.1; the solvable radical and the layer.... 59 4.1.4 Action of G on G=S ........................ 63 4.1.5 The factor group G=L is abelian-by-finite............. 63 4.1.6 The Frattini Argument....................... 64 4.1.7 Towards finiteness of G=L ..................... 65 4.1.8 Trimming the solvable radical................... 66 4.1.9 End of the proof.......................... 68 4.2 The definably simple case......................... 69 4.2.1 Proof of Theorem 4.2.1....................... 69 5 Groups of finite Morley rank 72 5.1 Introduction to groups of finite Morley rank............... 72 5.2 Infinite simple groups of finite Morley rank................ 77 5.3 Useful facts................................. 80 5.3.1 Classification of infinite simple locally finite groups of finite Morley rank............................. 80 5.3.2 Few rank tools........................... 81 5.3.3 Good tori.............................. 82 5.3.4 The Fitting subgroup and Borel subgroups............ 83 5.3.5 The socle.............................. 83 5.3.6 Two theorem's by Zilber...................... 84 5.3.7 Some Sylow 2-theory........................ 85 5.3.8 Strongly embedded subgroups................... 86 3 5.3.9 Some results on Frobenius groups and Zassenhaus groups of finite Morley rank.......................... 87 6 Finitary automorphism group A 89 6.1 The locally finite core............................ 89 6.2 Definably simple stable groups admitting A ............... 93 6.2.1 Proof of Theorem 6.2.1....................... 93 6.3 Groups of finite Morley rank admitting A ................. 95 6.3.1 Proof of Theorem 6.3.1....................... 95 7 The Hrushovski Programme for infinite simple groups of finite Morley rank 98 7.1 Results of U˘gurluand a tight automorphism α .............. 99 7.2 Small groups of finite Morley rank admitting α ............. 102 7.2.1 Our strategy............................. 103 7.2.2 Proof of Theorem 7.2.1....................... 106 7.3 Landscape for the future work....................... 123 Bibliography 129 Word count 30346 4 The University of Manchester Ulla Karhum¨aki Doctor of Philosophy On certain groups with finite centraliser dimension June 15, 2020 In this thesis, we study certain groups with finite centraliser dimension. These are groups in which there is a finite bound on the length of any chain of centralisers. We study locally finite groups of finite centraliser dimension and stable groups. In particular, we consider a special kind of stable groups called groups of finite Morley rank. The motivation to our work arises from two major conjectures in the topic of groups of finite Morley rank. One of them|the Cherlin{Zilber Conjecture|states that infinite simple groups of finite Morley rank are isomorphic to linear algebraic groups over algebraically closed fields. The other conjecture|the Principal Conjecture|is due to Ehud Hrushovski; it states that if an infinite simple group of finite Morley rank G admits a generic automorphism α, then the fixed point subgroup CG(α) is pseudofinite. It is known, by results of Zo´eChatzidakis and Hrushovski and results of Hrushovski alone, that the Cherlin{Zilber Conjecture implies the Principal Conjecture; some evidence suggests that the converse also holds. Chapters2,3 and5 contain no original results. In Chapters2 and3, we provide the necessary group-theoretic and model-theoretic background material required in later chapters. In Chapter5, we discuss groups of finite Morley rank and give a compendium of advanced results of the topic needed in Chapter7. Our results are proven in Chapters4,6 and7. Chapter4 is devoted to the study of locally finite groups of finite centraliser dimension. We prove a general result describing the structure of such groups. Moreover, we prove that definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. In Chapter6, we introduce a finitary automorphism group A. The definition of a finitary automorphism group A arises as we isolate certain properties of the group of Frobenius maps of an algebraically closed field K of positive characteristic. We prove that an infinite definably simple stable group admitting a finitary automorphism group A contains an infinite locally finite elementary subgroup. Then, we identify this locally finite elementary subgroup using our results in Chapter4. Consequently, we classify infinite definably simple stable groups admitting a finitary automorphism group A as Chevalley groups over algebraically closed fields of positive characteristic. In Chapter7, we continue the path designated by Pınar U˘gurlu. In her PhD thesis, U˘gurludeveloped a strategy towards proving the expected equivalence between the Cherlin{Zilber Conjecture and the Principal Conjecture. We first give U˘gurlu's definition of a tight automorphism α of an infinite simple group of finite Morley rank G. Then, we prove that, under suitable assumptions, a \small" infinite simple group of finite Morley rank admitting a tight automorphism α is isomorphic to P SL2(K) over some algebraically closed field K of characteristic different from 2. Keywords: Groups of finite centraliser dimension, locally finite groups, stable 5 groups, Frobenius maps, pseudofinite groups, groups of finite Morley rank, the Cherlin{ Zilber Conjecture, the Principal Conjecture 6 Declaration No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 7 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproductions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library's regulations (see http://www.manchester.ac.uk/library/ab- outus/regulations) and in The University's Policy on Presentation of Theses. 8 Acknowledgements It is impossible to thank my supervisor Alexandre Borovik enough. When I started my PhD, I was not very confident mathematically|quite the opposite. After four years of PhD under the supervision of Sasha, I have grown to be a mathematician who is not only interested but also not afraid to face new topics in mathematics. Sasha's patience guidance through these years allowed me to learn at my own phase; he always made me feel quite optimistic, even when I struggled to understand many things. I am thankful to Sasha for all his valuable ideas that allowed