DOCSLIB.ORG

Representation Growth

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Representation Growth Universidad Autónoma de Madrid Tesis Doctoral Representation Growth Autor: Director: Javier García Rodríguez Andrei Jaikin Zapirain Octubre 2016 a Laura. Abstract The main results in this thesis deal with the representation theory of certain classes of groups. More precisely, if rn(Γ) denotes the number of non-isomorphic n-dimensional complex representations of a group Γ, we study the numbers rn(Γ) and the relation of this arithmetic information with structural properties of Γ. In chapter 1 we present the required preliminary theory. In chapter 2 we introduce the Congruence Subgroup Problem for an algebraic group G defined over a global field k. In chapter 3 we consider Γ = G(OS) an arithmetic subgroup of a semisimple algebraic k-group for some global field k with ring of S-integers OS. If the Lie algebra of G is perfect, Lubotzky and Martin showed in [56] that if Γ has the weak Congruence Subgroup Property then Γ has Polynomial Representation Growth, that is, rn(Γ) ≤ p(n) for some polynomial p. By using a different approach, we show that the same holds for any semisimple algebraic group G including those with a non-perfect Lie algebra. In chapter 4 we apply our results on representation growth of groups of the form Γ = D log n G(OS) to show that if Γ has the weak Congruence Subgroup Property then sn(Γ) ≤ n for some constant D, where sn(Γ) denotes the number of subgroups of Γ of index at most n. As before, this extends similar results of Lubotzky [54], Nikolov, Abert, Szegedy [1] and Golsefidy [24] for almost simple groups with perfect Lie algebra to any simple algebraic k-group G. In chapter 5 we consider Γ = 1 + J, where J is a finite nilpotent associative algebra, this is called an algebra group. The Fake Degree Conjecture says that for algebra groups the numbers rn(Γ) may be obtained by looking at the square root of the sizes of the orbits arising from Kirillov’s Orbit Method. In [36] Jaikin gave a 2-group that served as a counterexample to this conjecture. We provide counterexamples for any prime p by looking at groups of the form Γ = 1 + IFq , where IFq is the augmentation ideal of the group algebra k(π)−1 Fq[π] for some p-group π. Moreover, we show that for such groups r1(Γ) = q | B0(π)|, where B0(π) is the Bogomolov multiplier of π. Q Finally in chapter 6, we consider Γ = i∈I Si, where the Si are nonabelian finite simple group. We give a characterization of the groups of this form having Polynomial Representation Growth. We also show that within this class one can obtain any rate of Q representation growth, i.e., for any α > 0 there exists Γ = i∈I Si where the Si are finite α simple groups of Lie type such that rn(Γ) ∼ n . Moreover, we may take all Si with a fixed Lie type. This complements results of Kassabov and Nikolov in [42]. i Agradecimientos Escribir 100 páginas en 4 años puede no parecer gran cosa. Confieso además que no lo habría podido hacer sin la ayuda de los que me han acompañado estos últimos años. Me gustaría darles las gracias por todas las líneas con las que, de una manera u otra, han contribuido. Gracias en primer lugar a Andrei por estar siempre dispuesto y de buen humor para trabajar, por la barbaridad de cosas que he aprendido y por el placer y la motivación que da trabajar con alguien con tanto talento. Gracias también a Joan por ejercer de hermano mayor. Por los ratos hablando de y pensando en grupos y sus cosas y por todos los buenos ratos que hemos pasado fuera de los grupos y sus cosas. Gracias a Urban porque los meses de trabajo y estudio juntos han sido los más satisfactorios. Gracias a Benjamin por acogerme en Düsseldorf y a toda la gente de allí que me hizo sentir como uno más durante mi estancia. La mejor parte de hacer el doctorado en la UAM ha sido sin duda el grupo de gente con el que he compartido estos años y que han conseguido que madrugara con ganas de irme a la otra punta de Madrid. Empezando por el principio, gracias a Felipe y a Juan por acogerme desde el primer día y ser los mejores tutores de UAM que uno podría tener. Gracias especiales a Ángel, compañero de aventuras desde el primer día y compañero de todos los despachos por los que he pasado. Gracias por tu buen humor, por las charlas matemáticas y no matemáticas y porque siempre apetece echar un café contigo. Gracias en general a toda la cantera de la UAM por aceptar a uno de la Complu sin rechistar (aunque siempre recordándole que la UAM es mejor y más difícil). Desde los veteranos Ana Z, Félix, David T., Bego, Sera, Jose, María, Adri, que ya se fueron a ligas mayores, los que han durado desde el principo hasta el final Carlos, David, Bea P., Felipe, Juan, Irina y los más prometedores Marcos, Bea B, Raquel, Dani, Marta... y también a los mejores fichajes Luis Daniel, Iason, Leyter y Ale. Desayunar, comer (mejor con café y discusión), merendar, cenar, beber, organizar regalos de tesis, celebrar tesis y lo que se terciara ha sido un placer. Mención especial merecen las tardes de fútbol y los Pi Warriors, siempre quedarán en mi memoria. Gracias por aguantarme a mis otros compañeros de despacho Carlos P., Jose Luis, Adri, Jaime y en especial a David por tener el récord de horas de despacho conmigo y servirme de avanzadilla en todo momento, te mereces una Champions (incluso ganándole la final al Madrid). Gracias a mis algebristas preferidos Carlos y Bea por ayudarme con mis eternas dudas de geometría algebraica. Perdón Carlos por las tardes que te he hecho perder y gracias por iii enseñarme con paciencia ese lenguaje tan raro que habláis. Gracias a mis analistas preferidos Jose y Adri por la no numerable cantidad de cosas que he aprendido de vosotros y sobre todo por convencerme de volver a la lucha armada. Gracias a mi matemática preferida, María Medina. Por empezar todas las mañanas con una gran sonrisa y hacer que un mal café sea el mejor de todos. Por reverdecer acelgas por muy pochas que estén, por todas las historias contadas, escritas y por escribir. Gracias a mi Jota, por estar desde siempre y para siempre. Por saber que tengo una amiga que no puedo perder por muchas tesis de 1000 páginas que tenga que hacer. Gracias a mis otros amigos matemáticos, Bruno y Poli, por todos los buenos momentos desde que empezamos a hacer cuentas. Gracias a Fer, el matemático que más me ha influenciado y acompañado desde que empecé, compañero de recomendaciones, amarguras y eternas discusiones, gracias por confiar siempre en mí, especialmente en los malos momentos. Gracias a toda mi familia por apoyarme en todas las etapas hasta llegar hasta aquí. En especial gracias a mis padres y a mi hermana por apoyarme siempre en todo lo que he querido hacer y ayudarme en todo lo que han podido. Gracias con cariño a mis abuelos, por estar siempre orgullosos incluso aunque no entendieran lo que hacía. Y gracias a ti Laura. Por acompañarme y ayudarme en todo desde siempre. Porque sin ti no habría podido, ni esto ni otras muchas cosas. Porque estás 100 páginas también son mérito, esfuerzo y sacrificio tuyo. Por ser la mejor parte de todo desde que tengo memoria. iv Contents Abstract i Agradecimientos iii Introducción 1 Introduction 7 1 Preliminaries 13 1.1 Global and local fields . 13 1.2 Topological groups . 15 1.2.1 Topological groups and group completions . 17 1.2.2 Profinite groups . 18 1.2.3 Analytic groups . 20 1.3 Affine algebraic varieties . 22 1.4 Algebraic groups . 26 1.5 Affine group schemes . 30 1.6 Representation Theory . 31 1.7 Finite groups of Lie type . 33 1.8 Commutative algebra . 33 2 The Congruence Subgroup Problem 35 2.1 Introduction . 35 2.2 Arithmetic groups . 36 2.3 The general formulation . 40 2.4 Reductions and necessary conditions . 41 2.5 The characteristic 0 case . 43 2.6 The semisimple case . 44 3 Representation Growth of Arithmetic Groups 47 3.1 Representation Growth . 47 3.2 CSP and Representation Growth of Arithmetic Groups . 55 v 4 Subgroup Growth of Arithmetic Groups 71 4.1 Subgroup Growth . 71 4.2 CSP and Subgroup Growth of Arithmetic Groups . 73 5 Algebra groups and the Fake Degree Conjecture 77 5.1 Introduction . 77 5.2 Algebra groups and Kirillov’s Orbit Method . 81 5.3 The Fake Degree Conjecture: a disproof . 87 5.4 Abelianization of groups of units and the Bogomolov multiplier . 90 5.4.1 The group (1 + IFq )ab .......................... 90 5.4.2 B0 and rationality questions in unipotent groups . 93 6 Finite groups of Lie type 97 6.1 Introduction . 97 6.2 Representation Growth of Finite Groups of Lie Type . 99 6.3 Polynomial Representation Growth in C .................... 100 6.4 Types of representation growth . 102 vi Introducción Sea Γ un grupo. Una representación de Γ es un homomorfismo de grupos ρ :Γ → GL(V ) para algún espacio vectorial complejo finito dimensional V . La Teoría de Representaciones estudia la relación entre propiedades estructurales de Γ y sus representaciones. El Teorema de Burnside fue la primera gran contribución de la Teoría de Representaciones y desde entonces se ha convertido en un área de investigación en sí misma.
Load more

Recommended publications
  • Complete Objects in Categories
    Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion.
    [Show full text]
  • Group Theory
    Appendix A Group Theory This appendix is a survey of only those topics in group theory that are needed to understand the composition of symmetry transformations and its consequences for fundamental physics. It is intended to be self-contained and covers those topics that are needed to follow the main text. Although in the end this appendix became quite long, a thorough understanding of group theory is possible only by consulting the appropriate literature in addition to this appendix. In order that this book not become too lengthy, proofs of theorems were largely omitted; again I refer to other monographs. From its very title, the book by H. Georgi [211] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [102] is written in the same spirit. Both books also cover the necessary group theory for grand unification ideas. A very comprehensive but also rather dense treatment is given by [428]. Still a classic is [254]; it contains more about the treatment of dynamical symmetries in quantum mechanics. A.1 Basics A.1.1 Definitions: Algebraic Structures From the structureless notion of a set, one can successively generate more and more algebraic structures. Those that play a prominent role in physics are defined in the following. Group A group G is a set with elements gi and an operation ◦ (called group multiplication) with the properties that (i) the operation is closed: gi ◦ g j ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi , (iii) for every gi exists an −1 ∈ ◦ −1 = = −1 ◦ inverse element gi G such that gi gi g0 gi gi , (iv) the operation is associative: gi ◦ (g j ◦ gk) = (gi ◦ g j ) ◦ gk.
    [Show full text]
  • 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
    [Show full text]
  • Projective Representations of Groups
    PROJECTIVE REPRESENTATIONS OF GROUPS EDUARDO MONTEIRO MENDONCA Abstract. We present an introduction to the basic concepts of projective representations of groups and representation groups, and discuss their relations with group cohomology. We conclude the text by discussing the projective representation theory of symmetric groups and its relation to Sergeev and Hecke-Clifford Superalgebras. Contents Introduction1 Acknowledgements2 1. Group cohomology2 1.1. Cohomology groups3 1.2. 2nd-Cohomology group4 2. Projective Representations6 2.1. Projective representation7 2.2. Schur multiplier and cohomology class9 2.3. Equivalent projective representations 10 3. Central Extensions 11 3.1. Central extension of a group 12 3.2. Central extensions and 2nd-cohomology group 13 4. Representation groups 18 4.1. Representation group 18 4.2. Representation groups and projective representations 21 4.3. Perfect groups 27 5. Symmetric group 30 5.1. Representation groups of symmetric groups 30 5.2. Digression on superalgebras 33 5.3. Sergeev and Hecke-Clifford superalgebras 36 References 37 Introduction The theory of group representations emerged as a tool for investigating the structure of a finite group and became one of the central areas of algebra, with important connections to several areas of study such as topology, Lie theory, and mathematical physics. Schur was Date: September 7, 2017. Key words and phrases. projective representation, group, symmetric group, central extension, group cohomology. 2 EDUARDO MONTEIRO MENDONCA the first to realize that, for many of these applications, a new kind of representation had to be introduced, namely, projective representations. The theory of projective representations involves homomorphisms into projective linear groups. Not only do such representations appear naturally in the study of representations of groups, their study showed to be of great importance in the study of quantum mechanics.
    [Show full text]
  • THE CONJUGACY CLASS RANKS of M24 Communicated by Robert Turner Curtis 1. Introduction M24 Is the Largest Mathieu Sporadic Simple
    International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 6 No. 4 (2017), pp. 53-58. ⃝c 2017 University of Isfahan ijgt.ui.ac.ir www.ui.ac.ir THE CONJUGACY CLASS RANKS OF M24 ZWELETHEMBA MPONO Dedicated to Professor Jamshid Moori on the occasion of his seventieth birthday Communicated by Robert Turner Curtis 10 3 Abstract. M24 is the largest Mathieu sporadic simple group of order 244823040 = 2 ·3 ·5·7·11·23 and contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of M24 with respect to the conjugacy classes of all its nonidentity elements. 1. Introduction 10 3 M24 is the largest Mathieu sporadic simple group of order 244823040 = 2 ·3 ·5·7·11·23 and contains all the other Mathieu sporadic simple groups as subgroups. It is a 5-transitive permutation group on a set of 24 points such that: (i) the stabilizer of a point is M23 which is 4-transitive (ii) the stabilizer of two points is M22 which is 3-transitive (iii) the stabilizer of a dodecad is M12 which is 5-transitive (iv) the stabilizer of a dodecad and a point is M11 which is 4-transitive M24 has a trivial Schur multiplier, a trivial outer automorphism group and it is the automorphism group of the Steiner system of type S(5,8,24) which is used to describe the Leech lattice on which M24 acts. M24 has nine conjugacy classes of maximal subgroups which are listed in [4], its ordinary character table is found in [4], its blocks and its Brauer character tables corresponding to the various primes dividing its order are found in [9] and [10] respectively and its complete prime spectrum is given by 2; 3; 5; 7; 11; 23.
    [Show full text]
  • Groups and Rings
    Groups and Rings David Pierce February , , : p.m. Matematik Bölümü Mimar Sinan Güzel Sanatlar Üniversitesi [email protected] http://mat.msgsu.edu.tr/~dpierce/ Groups and Rings This work is licensed under the Creative Commons Attribution–Noncommercial–Share-Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ CC BY: David Pierce $\ C Mathematics Department Mimar Sinan Fine Arts University Istanbul, Turkey http://mat.msgsu.edu.tr/~dpierce/ [email protected] Preface There have been several versions of the present text. The first draft was my record of the first semester of the gradu- ate course in algebra given at Middle East Technical University in Ankara in –. I had taught the same course also in –. The main reference for the course was Hungerford’s Algebra []. I revised my notes when teaching algebra a third time, in – . Here I started making some attempt to indicate how theorems were going to be used later. What is now §. (the development of the natural numbers from the Peano Axioms) was originally pre- pared for a course called Non-Standard Analysis, given at the Nesin Mathematics Village, Şirince, in the summer of . I built up the foundational Chapter around this section. Another revision, but only partial, came in preparation for a course at Mimar Sinan Fine Arts University in Istanbul in –. I expanded Chapter , out of a desire to give some indication of how mathematics, and especially algebra, could be built up from some simple axioms about the relation of membership—that is, from set theory.
    [Show full text]
  • The Group of Automorphisms of the Holomorph of a Group
    Pacific Journal of Mathematics THE GROUP OF AUTOMORPHISMS OF THE HOLOMORPH OF A GROUP NAI-CHAO HSU Vol. 11, No. 3 BadMonth 1961 THE GROUP OF AUTOMORPHISMS OF THE HOLOMORPH OF A GROUP NAI-CHAO HSU l Introduction* If G = HK where H is a normal subgroup of the group G and where K is a subgroup of G with the trivial intersection with H, then G is said to be a semi-direct product of H by K or a splitting extension of H by K. We can consider a splitting extension G as an ordered triple (H, K; Φ) where φ is a homomorphism of K into the automorphism group 2I(if) of H. The ordered triple (iϊ, K; φ) is the totality of all ordered pairs (h, k), he H, he K, with the multiplication If φ is a monomorphism of if into §I(if), then (if, if; φ) is isomorphic to (iϊ, Φ(K); c) where c is the identity mapping of φ(K), and therefore G is the relative holomorph of if with respect to a subgroup φ(-K) of Sί(ίf). If φ is an isomorphism of K onto Sί(iϊ), then G is the holomorph of if. Let if be a group, and let G be the holomorph of H. We are con- sidering if as a subgroup of G in the usual way. GoΓfand [1] studied the group Sί^(G) of automorphisms of G each of which maps H onto itself, the group $(G) of inner automorphisms of G, and the factor group SIff(G)/$5(G).
    [Show full text]
  • Minimum Circuit Size, Graph Isomorphism, and Related Problems∗
    SIAM J. COMPUT. c 2018 Society for Industrial and Applied Mathematics Vol. 47, No. 4, pp. 1339{1372 MINIMUM CIRCUIT SIZE, GRAPH ISOMORPHISM, AND RELATED PROBLEMS∗ ERIC ALLENDERy , JOSHUA A. GROCHOWz , DIETER VAN MELKEBEEKx , CRISTOPHER MOORE{, AND ANDREW MORGANx Abstract. We study the computational power of deciding whether a given truth table can be described by a circuit of a given size (the minimum circuit size problem, or MCSP for short) and of the variant denoted as MKTP, where circuit size is replaced by a polynomially related Kolmogorov measure. Prior to our work, all reductions from supposedly intractable problems to MCSP/MKTP hinged on the power of MCSP/MKTP to distinguish random distributions from distributions pro- duced by hardness-based pseudorandom generator constructions. We develop a fundamentally dif- ferent approach inspired by the well-known interactive proof system for the complement of graph isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include linear code equivalence, permutation group conjugacy, and matrix subspace conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest. Key words. reductions between NP-intermediate problems, graph isomorphism, minimum circuit size problem, time-bounded Kolmogorov complexity AMS subject classifications. 68Q15, 68Q17, 68Q30 DOI. 10.1137/17M1157970 1. Introduction. Finding a circuit of minimum size that computes a given Boolean function constitutes the overarching goal in nonuniform complexity theory.
    [Show full text]
  • View This Volume's Front and Back Matter
    http://dx.doi.org/10.1090/gsm/092 Finite Group Theory This page intentionally left blank Finit e Grou p Theor y I. Martin Isaacs Graduate Studies in Mathematics Volume 92 f//s ^ -w* American Mathematical Society ^ Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 20B15, 20B20, 20D06, 20D10, 20D15, 20D20, 20D25, 20D35, 20D45, 20E22, 20E36. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-92 Library of Congress Cataloging-in-Publication Data Isaacs, I. Martin, 1940- Finite group theory / I. Martin Isaacs. p. cm. — (Graduate studies in mathematics ; v. 92) Includes index. ISBN 978-0-8218-4344-4 (alk. paper) 1. Finite groups. 2. Group theory. I. Title. QA177.I835 2008 512'.23—dc22 2008011388 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. Reprinted with corrections by the American Mathematical Society, 2011.
    [Show full text]
  • On the Order of Schur Multipliers of Finite Abelian P-Groups
    Int. J. Contemp. Math. Sci., Vol. 2, 2007, no. 10, 479 - 485 On the Order of Schur Multipliers of Finite Abelian p-Groups B. Mashayekhy*, F. Mohammadzadeh and A. Hokmabadi Department of Mathematics Center of Excellence in Analysis on Algebraic Structures Ferdowsi University of Mashhad P.O. Box 1159-91775, Mashhad, Iran * Correspondence: [email protected] Abstract n n(n−1) −t Let G be a finite p-group of order p with |M(G)| = p 2 , where M(G) is the Schur multiplier of G. Ya.G. Berkovich, X. Zhou, and G. Ellis have determined the structure of G when t =0, 1, 2, 3. In this paper, we are going to find some structures for an abelian p-group G with conditions on the exponents of G, M(G), and S2M(G), where S2M(G) is the metabelian multiplier of G. Mathematics Subject Classification: 20C25; 20D15; 20E34; 20E10 Keywords: Schur multiplier, Abelian p- group 1 Introduction and Preliminaries ∼ Let G be any group with a presentation G = F/R, where F is a free group. Then the Baer invariant of G with respect to the variety of groups V, denoted by VM(G), is defined to be R ∩ V (F ) VM(G)= , [RV ∗F ] where V is the set of words of the variety V, V (F ) is the verbal subgroup of F and 480 B. Mashayekhy, F. Mohammadzadeh and A. Hokmabadi ∗ −1 [RV F ]=v(f1, ..., fi−1,fir, fi+1, ..., fn)v(f1, ..., fi, ..., fn) | r ∈ R, fi ∈ F, v ∈ V,1 ≤ i ≤ n, n ∈ N.
    [Show full text]
  • Notes on Finite Group Theory
    Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geome- try (where groups describe in a very detailed way the symmetries of geometric objects) and in the theory of polynomial equations (developed by Galois, who showed how to associate a finite group with any polynomial equation in such a way that the structure of the group encodes information about the process of solv- ing the equation). These notes are based on a Masters course I gave at Queen Mary, University of London. Of the two lecturers who preceded me, one had concentrated on finite soluble groups, the other on finite simple groups; I have tried to steer a middle course, while keeping finite groups as the focus. The notes do not in any sense form a textbook, even on finite group theory. Finite group theory has been enormously changed in the last few decades by the immense Classification of Finite Simple Groups. The most important structure theorem for finite groups is the Jordan–Holder¨ Theorem, which shows that any finite group is built up from finite simple groups. If the finite simple groups are the building blocks of finite group theory, then extension theory is the mortar that holds them together, so I have covered both of these topics in some detail: examples of simple groups are given (alternating groups and projective special linear groups), and extension theory (via factor sets) is developed for extensions of abelian groups. In a Masters course, it is not possible to assume that all the students have reached any given level of proficiency at group theory.
    [Show full text]
  • Brauer Groups of Linear Algebraic Groups
    Volume 71, Number 2, September 1978 BRAUERGROUPS OF LINEARALGEBRAIC GROUPS WITH CHARACTERS ANDY R. MAGID Abstract. Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d - l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, characterless, linear algebraic group over an algebraically closed field of characteristic zero: the Brauer group is finite-in fact, it is the Schur multiplier of the fundamental group of the algebraic group [6, Theorem 4.1, p. 299]. In this note, we extend these calculations to an arbitrary connected linear group in characteristic zero. The main result is the determination of the Brauer group of a ¿/-dimen- sional affine algebraic torus, which is shown to be (Q/Z)(n) where n = d(d — l)/2. (This result is noted in [6, 4.8, p. 301] when d = 2.) We then show that if G is a connected linear algebraic group whose character group has Z-rank d, then the Brauer group of G is C X (Q/Z)<n) where n is as above and C is finite. We adopt the following notational conventions: 7ms an algebraically closed field of characteristic zero, and T = (F*)id) is a ^-dimensional affine torus over F.
    [Show full text]