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Expansion in Finite Simple Groups of Lie Type
Expansion in finite simple groups of Lie type Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: [email protected] In memory of Garth Gaudry, who set me on the road Contents Preface ix Notation x Acknowledgments xi Chapter 1. Expansion in Cayley graphs 1 x1.1. Expander graphs: basic theory 2 x1.2. Expansion in Cayley graphs, and Kazhdan's property (T) 20 x1.3. Quasirandom groups 54 x1.4. The Balog-Szemer´edi-Gowers lemma, and the Bourgain- Gamburd expansion machine 81 x1.5. Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality 94 x1.6. Non-concentration in subgroups 127 x1.7. Sieving and expanders 135 Chapter 2. Related articles 157 x2.1. Cayley graphs and the algebra of groups 158 x2.2. The Lang-Weil bound 177 x2.3. The spectral theorem and its converses for unbounded self-adjoint operators 191 x2.4. Notes on Lie algebras 214 x2.5. Notes on groups of Lie type 252 Bibliography 277 Index 285 vii Preface Expander graphs are a remarkable type of graph (or more precisely, a family of graphs) on finite sets of vertices that manage to simultaneously be both sparse (low-degree) and \highly connected" at the same time. They enjoy very strong mixing properties: if one starts at a fixed vertex of an (two-sided) expander graph and randomly traverses its edges, then the distribution of one's location will converge exponentially fast to the uniform distribution. For this and many other reasons, expander graphs are useful in a wide variety of areas of both pure and applied mathematics. -
CHAPTER 4 Representations of Finite Groups of Lie Type Let Fq Be a Finite
CHAPTER 4 Representations of finite groups of Lie type Let Fq be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the Fq-rational points of a connected reductive group G defined over Fq. For example, if n is a positive integer GLn(Fq) and SLn(Fq) are finite groups of Lie type. 0 In Let J = , where In is the n × n identity matrix. Let −In 0 t Sp2n(Fq) = { g ∈ GL2n(Fq) | gJg = J }. Then Sp2n(Fq) is a symplectic group of rank n and is a finite group of Lie type. For G = GLn(Fq) or SLn(Fq) (and some other examples), the standard Borel subgroup B of G is the subgroup of G consisting of the upper triangular elements in G.A standard parabolic subgroup of G is a subgroup of G which contains the standard Borel subgroup B. If P is a standard parabolic subgroup of GLn(Fq), then there exists a partition (n1, . , nr) of n (a set of positive integers nj such that n1 + ··· + nr = n) such that P = P(n1,...,nr ) = M n N, where M ' GLn1 (Fq) × · · · × GLnr (Fq) has the form A 0 ··· 0 1 0 A2 ··· 0 M = | A ∈ GL ( ), 1 ≤ j ≤ r . . .. .. j nj Fq . 0 ··· 0 Ar and In1 ∗ · · · ∗ 0 In2 · · · ∗ N = , . .. .. . 0 ··· 0 Inr where ∗ denotes arbitary entries in Fq. The subgroup M is called a (standard) Levi sub- group of P , and N is called the unipotent radical of P . Note that the partition (1, 1,..., 1) corresponds to B and (n) corresponds to G. -
12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups
12.6 Further Topics on Simple groups 387 12.6 Further Topics on Simple Groups This Web Section has three parts (a), (b) and (c). Part (a) gives a brief descriptions of the 56 (isomorphism classes of) simple groups of order less than 106, part (b) provides a second proof of the simplicity of the linear groups Ln(q), and part (c) discusses an ingenious method for constructing a version of the Steiner system S(5, 6, 12) from which several versions of S(4, 5, 11), the system for M11, can be computed. 12.6(a) Simple Groups of Order less than 106 The table below and the notes on the following five pages lists the basic facts concerning the non-Abelian simple groups of order less than 106. Further details are given in the Atlas (1985), note that some of the most interesting and important groups, for example the Mathieu group M24, have orders in excess of 108 and in many cases considerably more. Simple Order Prime Schur Outer Min Simple Order Prime Schur Outer Min group factor multi. auto. simple or group factor multi. auto. simple or count group group N-group count group group N-group ? A5 60 4 C2 C2 m-s L2(73) 194472 7 C2 C2 m-s ? 2 A6 360 6 C6 C2 N-g L2(79) 246480 8 C2 C2 N-g A7 2520 7 C6 C2 N-g L2(64) 262080 11 hei C6 N-g ? A8 20160 10 C2 C2 - L2(81) 265680 10 C2 C2 × C4 N-g A9 181440 12 C2 C2 - L2(83) 285852 6 C2 C2 m-s ? L2(4) 60 4 C2 C2 m-s L2(89) 352440 8 C2 C2 N-g ? L2(5) 60 4 C2 C2 m-s L2(97) 456288 9 C2 C2 m-s ? L2(7) 168 5 C2 C2 m-s L2(101) 515100 7 C2 C2 N-g ? 2 L2(9) 360 6 C6 C2 N-g L2(103) 546312 7 C2 C2 m-s L2(8) 504 6 C2 C3 m-s -
Contents 1 Root Systems
Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations. -
The Mathieu Groups (Simple Sporadic Symmetries)
The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 1 / 15 The Mathieu Groups (Simple Sporadic Symmetries) Scott Harper (University of St Andrews) Tomorrow's Mathematicians Today 21st February 2015 Scott Harper The Mathieu Groups 21st February 2015 2 / 15 1 2 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the 4 3 symmetry group of the object. Symmetry group: D4 The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A symmetry is a structure preserving permutation of the underlying set. A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. ∼ h(1 2 3 4)i = C4 Subgroup fixing 1: h(2 4)i Symmetry 1 2 4 3 Symmetry group: D4 Scott Harper The Mathieu Groups 21st February 2015 3 / 15 A group acts faithfully on an object if it is isomorphic to a subgroup of the symmetry group of the object. The stabiliser of a point in a group G is Group of rotations: the subgroup of G which fixes x. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. -
The Purpose of These Notes Is to Explain the Outer Automorphism Of
The Goal: The purpose of these notes is to explain the outer automorphism of the symmetric group S6 and its connection to the Steiner (5, 6, 12) design and the Mathieu group M12. In view of giving short proofs, we often appeal to a mixture of logic and direct computer-assisted enumerations. Preliminaries: It is an exercise to show that the symmetric group Sn ad- mits an outer automorphism if and only if n = 6. The basic idea is that any automorphism of Sn must map the set of transpositions bijectively to the set of conjugacy classes of a single permutation. A counting argument shows that the only possibility, for n =6 6, is that the automorphism maps the set of transpositions to the set of transpositions. From here it is not to hard to see that the automorphism must be an inner automorphism – i.e., conjugation by some element of Sn. For n = 6 there are 15 transpositions and also 15 triple transpositions, permutations of the form (ab)(cd)(ef) with all letters distinct. This leaves open the possibility of an outer automorphism. The Steiner Design: Let P1(Z/11) denote the projective line over Z/11. The group Γ = SL2(Z/11) consists of those transformations x → (ax + b)/(cx + d) with ad − bc ≡ 1 mod 11. There are 660 = (12 × 11 × 10)/2 elements in this group. Let B∞ = {1, 3, 4, 5, 9, ∞} ⊂ P1(Z/11) be the 6-element subset consisting of the squares mod 11 and the element ∞. Let Γ∞ ⊂ Γ be the stabilizer of B∞. -
2020 Ural Workshop on Group Theory and Combinatorics
Institute of Natural Sciences and Mathematics of the Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center 2020 Ural Workshop on Group Theory and Combinatorics Yekaterinburg – Online, Russia, August 24-30, 2020 Abstracts Yekaterinburg 2020 2020 Ural Workshop on Group Theory and Combinatorics: Abstracts of 2020 Ural Workshop on Group Theory and Combinatorics. Yekaterinburg: N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 2020. DOI Editor in Chief Natalia Maslova Editors Vladislav Kabanov Anatoly Kondrat’ev Managing Editors Nikolai Minigulov Kristina Ilenko Booklet Cover Desiner Natalia Maslova c Institute of Natural Sciences and Mathematics of Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center, 2020 2020 Ural Workshop on Group Theory and Combinatorics Conents Contents Conference program 5 Plenary Talks 8 Bailey R. A., Latin cubes . 9 Cameron P. J., From de Bruijn graphs to automorphisms of the shift . 10 Gorshkov I. B., On Thompson’s conjecture for finite simple groups . 11 Ito T., The Weisfeiler–Leman stabilization revisited from the viewpoint of Terwilliger algebras . 12 Ivanov A. A., Densely embedded subgraphs in locally projective graphs . 13 Kabanov V. V., On strongly Deza graphs . 14 Khachay M. Yu., Efficient approximation of vehicle routing problems in metrics of a fixed doubling dimension . -
Polytopes Derived from Sporadic Simple Groups
Volume 5, Number 2, Pages 106{118 ISSN 1715-0868 POLYTOPES DERIVED FROM SPORADIC SIMPLE GROUPS MICHAEL I. HARTLEY AND ALEXANDER HULPKE Abstract. In this article, certain of the sporadic simple groups are analysed, and the polytopes having these groups as automorphism groups are characterised. The sporadic groups considered include all with or- der less than 4030387201, that is, all up to and including the order of the Held group. Four of these simple groups yield no polytopes, and the highest ranked polytopes are four rank 5 polytopes each from the Higman-Sims group, and the Mathieu group M24. 1. Introduction The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if 2 0 the Tits group F4(2) is counted also) which these infinite families do not include. These sporadic simple groups range in size from the Mathieu group 53 M11 of order 7920, to the Monster group M of order approximately 8×10 . One key to the study of these groups is identifying some geometric structure on which they act. This provides intuitive insight into the structure of the group. Abstract regular polytopes are combinatorial structures that have their roots deep in geometry, and so potentially also lend themselves to this purpose. Furthermore, if a polytope is found that has a sporadic group as its automorphism group, this gives (in theory) a presentation of the sporadic group over some generating involutions. In [6], a simple algorithm was given to find all polytopes acted on by a given abstract group. -
Subgroups of Finite Groups of Lie Type
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 61, 16-27 (1979) Subgroups of Finite Groups of Lie Type GARY R/I. SEITZ* Urrivrrsity of Oregon, Eugene, Oregon Communicated by Walter F&t Received January 9, 1979 I. INTRODUCTION Let G :- G(p) be a finite group of Lie type defined over the field F, . Choose a Bore1 subgroup, B = UH .< G, of G, where Zi is unipotent and H a Cartan subgroup of G. In this paper we are concerned with the subgroups, Y, of G, such that 11 :< Y, and we determine these subgroups in the case where q is odd andg > 11. Associated with G is a root system, z, and a collection of root subgroups {U, : a: E Z> such that C’ 1 nI,,,- Zrz and such that FI < N(U,) for- each a E ,X In Lemma 3 of [ 1 I] it was shown that for q :‘-- 4 any H-invariant subgroup of U is essentially a product of root subgroups (the word “essentially” is relevant only when G is twisted, with some root subgroup non-Abelian). This I-es& was cxtcnded in [7], where it was shown that any unipotent subgroup of G normalized by H is of this form (although now negative roots are allowed). THEOREM. Suppose q is odd and q :> 11. Let H << t’ :$ G and set k; (ZTaCI YlatzZ\. Then (i) I’,, 4 I- and Y = 17,,A~,.(H); (ii) Y,, = : UOXO , where U, f3 LYO _ I, L), is u&potent and X0 is a central product of groups of Lie type; (iii) Zr,, and each component of-Y,, is generated b?l groups of the.form lJa n Y, 01E 2’; and (iv) If G f ‘G,(q), then for a E Z, lTa TI Y = 1, .L, , or Q(Q). -
Final Report (PDF)
GROUPS AND GEOMETRIES Inna Capdeboscq (Warwick, UK) Martin Liebeck (Imperial College London, UK) Bernhard Muhlherr¨ (Giessen, Germany) May 3 – 8, 2015 1 Overview of the field, and recent developments As groups are just the mathematical way to investigate symmetries, it is not surprising that a significant number of problems from various areas of mathematics can be translated into specialized problems about permutation groups, linear groups, algebraic groups, and so on. In order to go about solving these problems a good understanding of the finite and algebraic groups, especially the simple ones, is necessary. Examples of this procedure can be found in questions arising from algebraic geometry, in applications to the study of algebraic curves, in communication theory, in arithmetic groups, model theory, computational algebra and random walks in Markov theory. Hence it is important to improve our understanding of groups in order to be able to answer the questions raised by all these areas of application. The research areas covered at the meeting fall into three main inter-related topics. 1.1 Fusion systems and finite simple groups The subject of fusion systems has its origins in representation theory, but it has recently become a fast growing area within group theory. The notion was originally introduced in the work of L. Puig in the late 1970s; Puig later formalized this concept and provided a category-theoretical definition of fusion systems. He was drawn to create this new tool in part because of his interest in the work of Alperin and Broue,´ and modular representation theory was its first berth. It was then used in the field of homotopy theory to derive results about the p-completed classifying spaces of finite groups. -
On the Lie Algebra of a Finite Group of Lie Type
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 74, 333-340 (1982) On the Lie Algebra of a Finite Group of Lie Type MASAHARU KANEDA AND GARY M. SEITZ* Department of Mathematics, University of Oregon, Eugene, Oregon 97403 Communicated by Walter Feit Received October 18, 1980 0. INTRODUCTION In this paper we study a certain finite Lie algebra, g, associated with a finite group of Lie type, G. The maximal tori of g are determined (assuming certain characteristic restrictions), and it is shown that the number of G- orbits of these depends only on the Weyl group. For each maximal torus of g there is a decomposition of g into generalized weight spaces. With additional hypotheses we show that each subgroup of G containing a maximal torus has an associated subalgebra of g, and the subalgebras arising in this manner are precisely those that contain a maximal torus. The association provides a computationally effective linearization of the main results of [3]. The proofs of the above results are fairly easy given the existing literature on modular Lie algebras, standard results on maximal tori in G, and results from [3]. However, we note that the required group theoretic results are in fact based on the full classification of finite simple groups. 1. NOTATION AND BASIC LEMMAS Fix a prime p > 5 and G a simple algebraic group over IF,,. Let u be a surjective endomorphism of G such that G = G,, is a finite group of Lie type.