ON CERTAIN GROUPS WITH FINITE CENTRALISER DIMENSION

Abstract

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, then the fixed point subgroup of this generic automorphism is pseudofinite. It is known, by results of Zoe Chatzidakis and Hrushovski and results of Hrushovski alone, that the Cherlin-Zilber Conjecture implies the Principal Conjecture; some evidence suggests that the converse also holds. Chapters 2, 3 and 5 contain no original results. In Chapters 2 and 3, we provide the necessary group-theoretic and model-theoretic background material required in later chapters. In Chapter 5, we discuss groups of finite Morley rank and give a compendium of advanced results of the topic needed in Chapter 7. Our results are proven in Chapters 4, 6 and 7. Chapter 4 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 Chapter 6, 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 Chapter 4. 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 Chapter 7, we continue the path designated by Pinar Ugurlu. In her PhD thesis, Ugurlu developed a strategy towards proving the expected equivalence between the Cherlin-Zilber Conjecture and the Principal Conjecture. We first give Ugurlu's definition of a tight automorphism 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 is isomorphic to PSL_2(K) over some algebraically closed field K of characteristic different from 2.

Date of Award	1 Aug 2020
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Alexandre Borovik (Supervisor) & Hendrik Süß (Supervisor)