Applications of Lie Ring Methods to Group Theory
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Discrete and Profinite Groups Acting on Regular Rooted Trees
Discrete and Profinite Groups Acting on Regular Rooted Trees Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen vorgelegt von Olivier Siegenthaler aus Lausanne G¨ottingen, den 31. August 2009 Referent: Prof. Dr. Laurent Bartholdi Korreferent: Prof. Dr. Thomas Schick Tag der m¨undlichen Pr¨ufung: den 28. September 2009 Contents Introduction 1 1 Foundations 5 1.1 Definition of Aut X∗ and ...................... 5 A∗ 1.2 ZariskiTopology ............................ 7 1.3 Actions of X∗ .............................. 8 1.4 Self-SimilarityandBranching . 9 1.5 Decompositions and Generators of Aut X∗ and ......... 11 A∗ 1.6 The Permutation Modules k X and k X ............ 13 { } {{ }} 1.7 Subgroups of Aut X∗ .......................... 14 1.8 RegularBranchGroups . 16 1.9 Self-Similarity and Branching Simultaneously . 18 1.10Questions ................................ 19 ∗ 2 The Special Case Autp X 23 2.1 Definition ................................ 23 ∗ 2.2 Subgroups of Autp X ......................... 24 2.3 NiceGeneratingSets. 26 2.4 Uniseriality ............................... 28 2.5 Signature and Maximal Subgroups . 30 2.6 Torsion-FreeGroups . 31 2.7 Some Specific Classes of Automorphisms . 32 2.8 TorsionGroups ............................. 34 3 Wreath Product of Affine Group Schemes 37 3.1 AffineSchemes ............................. 38 3.2 ExponentialObjects . 40 3.3 Some Hopf Algebra Constructions . 43 3.4 Group Schemes Corresponding to Aut X∗ .............. 45 3.5 Iterated Wreath Product of the Frobenius Kernel . 46 i Contents 4 Central Series and Automorphism Towers 49 4.1 Notation................................. 49 4.2 CentralSeries.............................. 51 4.3 Automorphism and Normalizer Towers . 54 5 Hausdorff Dimension 57 5.1 Definition ................................ 57 5.2 Layers .................................. 58 5.3 ComputingDimensions . -
On the Centralizer of an Element of Order Four in a Locally Finite Group
Glasgow Math. J. 49 (2007) 411–415. C 2007 Glasgow Mathematical Journal Trust. doi:10.1017/S0017089507003746. Printed in the United Kingdom ON THE CENTRALIZER OF AN ELEMENT OF ORDER FOUR IN A LOCALLY FINITE GROUP PAVEL SHUMYATSKY Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil e-mail: [email protected] (Received 12 December, 2006; revised 6 March, 2007; accepted 10 March, 2007) Abstract. We prove that if G is a locally finite group admitting an automorphism φ of order four such that CG(φ) is Chernikov, then G has a soluble subgroup of finite index. 1. Introduction. The study of centralizers in locally finite groups is an interesting direction of group theory. In particular, the situation that a locally finite group G admits an automorphism φ of finite order (or, equivalenly, contains an element φ) such that CG(φ) is Chernikov has received some attention in the past. Recall that a group C is Chernikov if it has a subgroup of finite index that is a direct product of finitely many groups of type Cp∞ for various primes p (quasicyclic p-groups). By a deep result obtained independently by Shunkov [22] and also by Kegel and Wehrfritz [7] Chernikov groups are precisely the locally finite groups satisfying the minimal condition on subgroups; that is, any non-empty set of subgroups possesses a minimal subgroup. Hartley proved in 1988 that if φ is of prime-power order, then G has a locally soluble subgroup of finite index (see [4, Theorem 1]). The case of the theorem where φ has order two was handled by Asar in [1] and for automorphisms of arbitrary prime order the theorem was proved by Turau in [25]. -
Abstract Quotients of Profinite Groups, After Nikolov and Segal
ABSTRACT QUOTIENTS OF PROFINITE GROUPS, AFTER NIKOLOV AND SEGAL BENJAMIN KLOPSCH Abstract. In this expanded account of a talk given at the Oberwolfach Ar- beitsgemeinschaft “Totally Disconnected Groups”, October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is [17]. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject. 1. Introduction §1.1. Many concepts and techniques in the theory of finite groups depend in- trinsically on the assumption that the groups considered are a priori finite. The theoretical framework based on such methods has led to marvellous achievements, including – as a particular highlight – the classification of all finite simple groups. Notwithstanding, the same methods are only of limited use in the study of infinite groups: it remains mysterious how one could possibly pin down the structure of a general infinite group in a systematic way. Significantly more can be said if such a group comes equipped with additional information, such as a structure-preserving action on a notable geometric object. A coherent approach to studying restricted classes of infinite groups is found by imposing suitable ‘finiteness conditions’, i.e., conditions that generalise the notion of being finite but are significantly more flexible, such as the group being finitely generated or compact with respect to a natural topology. One rather fruitful theme, fusing methods from finite and infinite group theory, consists in studying the interplay between an infinite group Γ and the collection of all its finite quotients. -
Groups with Bounded Centralizer Chains and The~ Borovik--Khukhro
Groups with bounded centralizer chains and the Borovik–Khukhro conjecture A. A. Buturlakin1,2,3, D. O. Revin1,2, A. V. Vasil′ev1,2 Abstract Let G be a locally finite group and F (G) the Hirsch–Plotkin radical of G. Denote by S the full inverse image of the generalized Fitting subgroup of G/F (G) in G. Assume that there is a number k such that the length of every chain of nested centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption the quotient G/S contains an abelian subgroup of index bounded in terms of k. We disprove this statement and prove some its weaker analog. Keywords: locally finite groups, centralizer lattice, c-dimension, k-tuple Dick- son’s conjecture Introduction Following [24], the maximal length of chains of nested centralizers of a group G is called the c-dimension of G (the same number is also known as the length of the centralizer lattice of G, see, e.g., [25]). Here we consider groups of finite c-dimension. The class of such groups is quite wide: it includes, for example, abelian groups, linear groups, torsion-free hyperbolic groups, and hence free groups. It is closed under taking subgroups, finite direct products, and extensions by finite groups; however, the c-dimension of a homomorphic image of a group from this class can be infinite. Obviously, each group of finite c-dimension satisfies the minimal condition on centralizers, therefore, it is a so-called MC -group. The minimal condition on centralizers is a useful notion in the class of arXiv:1708.03057v1 [math.GR] 10 Aug 2017 locally finite groups. -
Burnside, March 2016
A Straightforward Solution to Burnside’s Problem S. Bachmuth 1. Introduction The Burnside Problem for groups asks whether a finitely generated group, all of whose elements have bounded order, is finite. We present a straightforward proof showing that the 2-generator Burnside groups of prime power exponent are solvable and therefore finite. This proof is straightforward in that it does not rely on induced maps as in [2], but it is strongly dependent on Theorem B in the joint paper with H. A. Heilbronn and H. Y. Mochizuki [9]. Theorem B is reformulated here as Lemma 3(i) in Section 2. Our use of Lemma 3(i) is indispensable. Throughout this paper we fix a prime power q = pe and unless specifically mentioned otherwise, all groups are 2-generator. At appropriate places, we may require e = 1 so that q = p is prime; otherwise q may be any fixed prime power. The only (published) positive results of finiteness of Burnside groups of prime power exponents are for exponents q = 2, 3 and 4 ([10],[12]). Some authors, beginning with P. S. Novikoff and S. I. Adian {15], (see also [1}), have claimed that groups of exponent k are infinite for k sufficiently large. Our result here, as in [2], is at odds with this claim. Since this proof avoids the use of induced maps, Section 4 of [2] has been rewritten. Sections 2, 3 and 6 have been left unaltered apart from minor, mostly expository, changes and renumbering of items. The introduction and Section 5 have been rewritten. Sections 5 & 6 are not involved in the proof although Section 5 is strongly recommended. -
Uniformly Recurrent Subgroups of Simple Locally Finite Groups
UNIFORMLY RECURRENT SUBGROUPS OF SIMPLE LOCALLY FINITE GROUPS SIMON THOMAS Abstract. We study the uniformly recurrent subgroups of simple locally finite groups. 1. Introduction Let G be a countably infinite group and let SubG be the compact space of subgroups H 6 G. Then G acts as a group of homeomorphisms of SubG via the g −1 conjugation action, H 7! gHg . Following Glasner-Weiss [5], a subset X ⊆ SubG is said to be a uniformly recurrent subgroup or URS if X is a minimal G-invariant closed subset of SubG. For example, if N E G is a normal subgroup, then the singleton set f N g is a URS of G. Throughout this paper, these singleton URSs will be regarded as trivial URSs. More interesting examples of URSs arise as the stabilizer URSs of minimal actions. For example, suppose that ∆ is a compact space and that G y ∆ is a minimal G-action. Let f : ∆ ! SubG be the G-equivariant stabilizer map defined by x 7! Gx = f g 2 G j g · x = x g: and let X∆ = f(∆). If f is continuous, then it follows easily that X∆ is a URS of G; and, as expected, X∆ is called the stabilizer URS of the minimal action G y ∆. (It is well-known and easily checked that the map f : ∆ ! SubG is continuous if and only Fix∆(g) = f x 2 ∆ j g · x = x g is clopen for every g 2 G.) By Matte-Bon- Tsankov [16], if X ⊆ SubG is any URS of G, then there exists a minimal action G y ∆ such that the stabilizer map f : ∆ ! SubG is continuous and f(∆) = X. -
Congruence Subgroup Problem for Algebraic Groups: Old and New Astérisque, Tome 209 (1992), P
Astérisque A. S. RAPINCHUK Congruence subgroup problem for algebraic groups: old and new Astérisque, tome 209 (1992), p. 73-84 <http://www.numdam.org/item?id=AST_1992__209__73_0> © Société mathématique de France, 1992, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CONGRUENCE SUBGROUP PROBLEM FOR ALGEBRAIC GROUPS: OLD AND NEW A. S. RAPINCHUK* Let G C GLn be an algebraic group defined over an algebraic number field K. Let 5 be a finite subset of the set VK of all valuations of K, containing the set V*£ of archimedean valuations. Denote by O(S) the ring of 5-integers in K and by GQ(S) the group of 5-units in G. To any nonzero ideal a C O(S) there corresponds the congruence subgroup Go(s)(*) = {96 G0(s) \ 9 = En (mod a)} , which is a normal subgroup of finite index in GQ(S)- The initial statement of the Congruence Subgroup Problem was : (1) Does any normal subgroup of finite index in GQ(S) contain a suitable congruence subgroup Go(s)(a) ? In fact, it was found by F. Klein as far back as 1880 that for the group SL2(Z) the answer to question (1) is "no". -
Master's Thesis
MASTER'S THESIS Title of the Master's Thesis Engel Lie algebras submitted by Thimo Maria Kasper, BSc in partial fulfilment of the requirements for the degree of Master of Science (MSc) Vienna, 2018 Degree programme code: A 066 821 Degree programme: Master Mathematics Supervisor: Prof. Dr. Dietrich Burde Summary The purpose of this thesis is to present an investigation of Engel-n Lie algebras. In addition to the defining relations of Lie algebras these satisfy the so-called Engel-n identity ad(x)n = 0 for all x. Engel Lie algebras arise in the study of the Restricted Burnside Problem, which was solved by Efim Zelmanov in 1991. Beside a general introduction to the topic, special interest is taken in the exploration of the nilpotency classes of Engel-n Lie algebras for small values of n. At this, the primary objective is to elaborate the case of n = 3 explicitly. Chapter 1 concerns the general theory of Lie algebras. In the course of this, the essential properties of solvability and nilpotency are explained as they will be central in the subsequent discussion. Further, the definition of free Lie algebras which contributes to the establishment of the concept of free-nilpotent Lie algebras. In the last section several notions of group theory are surveyed. These will be useful in Chapter 2. The second chapter explains the origin and solution of the Burnside Problems. In that process, a historical survey on William Burnside and the first results on his fundamental questions are given. Next, the so-called Restricted Burnside Problem is considered and an overview of the most important steps to the solution is dis- played. -
Current Trends and Open Problems in Arithmetic Dynamics
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 56, Number 4, October 2019, Pages 611–685 https://doi.org/10.1090/bull/1665 Article electronically published on March 1, 2019 CURRENT TRENDS AND OPEN PROBLEMS IN ARITHMETIC DYNAMICS ROBERT BENEDETTO, PATRICK INGRAM, RAFE JONES, MICHELLE MANES, JOSEPH H. SILVERMAN, AND THOMAS J. TUCKER Abstract. Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geom- etry and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. Contents 1. Introduction 612 2. Abstract dynamical systems 613 3. Background: Number theory and algebraic geometry 615 4. Uniform boundedness of (pre)periodic points 617 5. Arboreal representations 619 6. Dynatomic representations 622 7. Intersections of orbits and subvarieties 624 8. (Pre)periodic points on subvarieties 626 9. Dynamical (dynatomic) modular curves 627 10. Dynamical moduli spaces 630 11. Unlikely intersections in dynamics 634 12. Good reduction of maps and orbits 636 13. Dynamical degrees of rational maps 642 14. Arithmetic degrees of orbits 645 15. Canonical heights 649 16. Variation of the canonical height 653 17. p-adic and non-archimedean dynamics 656 18. Dynamics over finite fields 661 19. Irreducibilty and stability of iterates 665 20. Primes, prime divisors, and primitive divisors in orbits 668 21. Integral points in orbits 671 Received by the editors June 30, 2018. -
On Certain Groups with Finite Centraliser Dimension
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...................... -
The CI Problem for Infinite Groups
The CI problem for infinite groups Joy Morris∗ Department of Mathematics and Computer Science University of Lethbridge Lethbridge, AB. T1K 3M4. Canada [email protected] Submitted: Feb 24, 2015; Accepted: Nov 21, 2016; Published: Dec 9, 2016 Mathematics Subject Classifications: 05C25, 20B27 Abstract A finite group G is a DCI-group if, whenever S and S0 are subsets of G with the Cayley graphs Cay(G; S) and Cay(G; S0) isomorphic, there exists an automorphism ' of G with '(S) = S0. It is a CI-group if this condition holds under the restricted assumption that S = S−1. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CIf -group if the same condition holds under the restricted assumption that S is finite; and an infinite group is a (D)CIf -group if the same condition holds whenever S is both finite and generates G. We prove that an infinite (D)CI-group must be a torsion group that is not locally- finite. We find infinite families of groups that are (D)CIf -groups but not strongly (D)CIf -groups, and that are strongly (D)CIf -groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely n characterise the locally-finite DCI-graphs on Z . We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CI- group exists. 1 Introduction Although there has been considerable work done on the Cayley Isomorphism problem for finite groups and graphs, little attention has been paid to its extension to the infinite case. -
Uncountable Universal Locally Finite Groups
JOURNAL OF ALGEBRA 43, 168-175 (1976) Uncountable Universal Locally Finite Groups ANGUS MACINTYRE* AND SAHARON SHELAH Department of Mathematics, Yale University, New Haven, Connecticut 06520 Department of Mathematics, The Hebrew University, Jerusalem, Israel Communicated by Walter Feit Received June 18, 1975 In this paper we solve some group-theoretic problems by the method of indiscernibles in infinitary logic. The problems under consideration come from the book “Locally Finite Groups,” by Kegel and Wehrfritz [3], and concern a class of groups first studied by Philip Hall [2]. Hall introduced the notion of universal locally finite group, and proved the following results: (a) Any locally finite group H of cardinal K can be embedded in a universal locally finite group G of cardinal maX(K, N,); (b) Any two countable universal locally finite groups are isomorphic. In the light of this Kegel and Wehrfritz raised two questions. [3, pp. 182-1831. These are I. SUppOSe K > 8,. Are any two universal locally finite groups of cardinal K isomorphic ? II. Suppose H is a locally finite group of cardinal K, where K > N, . Is H embeddable in every universal locally finite group of cardinal K ? We answer both questions in the negative for all uncountable K. Concerning I, we show that there are 2” isomorphism types of universal locally finite groups of cardinal K, if K > N, . Moreover, if K is regular and K > N, , there are 2” pairwise nonembeddable universal locally finite groups of cardinal K. Concerning II, we show, for example, that if K = 2*o then a counterexample is given by taking H = Ssw.