ELEMENTARY ABELIAN P-SUBGROUPS of ALGEBRAIC GROUPS
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REPRESENTATIONS of ELEMENTARY ABELIAN P-GROUPS and BUNDLES on GRASSMANNIANS
REPRESENTATIONS OF ELEMENTARY ABELIAN p-GROUPS AND BUNDLES ON GRASSMANNIANS JON F. CARLSON∗, ERIC M. FRIEDLANDER∗∗ AND JULIA PEVTSOVA∗∗∗ Abstract. We initiate the study of representations of elementary abelian p- groups via restrictions to truncated polynomial subalgebras of the group alge- p p bra generated by r nilpotent elements, k[t1; : : : ; tr]=(t1; : : : ; tr ). We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several gen- eral constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples. Contents 1. The r-rank variety Grass(r; V)M 4 2. Radicals and Socles 10 3. Modules of constant radical and socle rank 15 4. Modules from quantum complete intersections 20 5. Radicals of Lζ -modules 27 6. Construction of Bundles on Grass(r; V) 35 6.1. A local construction of bundles 36 6.2. A construction by equivariant descent 38 7. Bundles for GLn-equivariant modules. 43 8. A construction using the Pl¨ucker embedding 51 9. APPENDIX (by J. Carlson). Computing nonminimal 2-socle support varieties using MAGMA 57 References 59 Quillen's fundamental ideas on applying geometry to the study of group coho- mology in positive characteristic [Quillen71] opened the door to many exciting de- velopments in both cohomology and modular representation theory. Cyclic shifted subgroups, the prototypes of the rank r shifted subgroups studied in this paper, were introduced by Dade in [Dade78] and quickly became the subject of an intense study. -
Fusion Systems and Group Actions with Abelian Isotropy Subgroups
FUSION SYSTEMS AND GROUP ACTIONS WITH ABELIAN ISOTROPY SUBGROUPS OZG¨ UN¨ UNL¨ U¨ AND ERGUN¨ YALC¸IN Abstract. We prove that if a finite group G acts smoothly on a manifold M so that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly n1 n on M × S × · · · × S k for some positive integers n1; : : : ; nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres with trivial action on homology. 1. Introduction and statement of results Madsen-Thomas-Wall [7] proved that a finite group G acts freely on a sphere if (i) G has no subgroup isomorphic to the elementary abelian group Z=p × Z=p for any prime number p and (ii) has no subgroup isomorphic to the dihedral group D2p of order 2p for any odd prime number p. At that time the necessity of these conditions were already known: P. A. Smith [12] proved that the condition (i) is necessary and Milnor [8] showed the necessity of the condition (ii). As a generalization of the above problem we are interested in the problem of character- izing those finite groups which can act freely and smoothly on a product of k spheres for a given positive integer k. In the case k = 1, Madsen-Thomas-Wall uses surgery theory and considers the unit spheres of linear representations of subgroups to show that certain surgery obstructions vanish. -
Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere
Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere By Sacha Breton, B.Sc. A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of Master of Science McMaster University c Copyright by Sacha Breton, August 30th 2011 Master of Science (2011) McMaster University Hamilton, Ontario Department of Mathematics and Statistics TITLE: Finite Group Actions on the Four-Dimensional Sphere AUTHOR: Sacha Breton B.Sc. (McGill University) SUPERVISOR: Ian Hambleton NUMBER OF PAGES: 41 + v ii Abstract Smith theory provides powerful tools for understanding the geometry of singular sets of group actions on spheres. In this thesis, tools from Smith theory and spectral sequences are brought together to study the singular sets of elementary abelian groups acting locally linearly on S4. It is shown that the singular sets of such actions are homeomorphic to the singular sets of linear actions. A short review of the literature on group actions on S4 is included. iii Acknowledgements I would like to thank Professor Ian Hambleton for his suggestion of thesis topic, for the breadth and depth of his advice, and for his patience and encouragement. I would also like to thank Val´erieTremblay, Barnaby and Sue Ore, my parents, and my friends for their continued support. iv Contents Descriptive Note ii Abstract iii Acknowledgements iv 1 Introduction 1 2 Introductory Material 1 2.1 Basic definitions and conventions . .1 2.2 The Projection and the Transfer homomorphisms . .4 2.3 Smith Theory . .7 2.4 Group cohomology . -
Arxiv:1310.6548V1 [Math.GR] 24 Oct 2013 Every Hwta H Iuto O Ocmatmnflsi Ut Di Quite Is 3
FINITE SUBGROUPS OF DIFFEOMORPHISM GROUPS ∗ VLADIMIR L. POPOV Abstract. We prove (1) the existence of a noncompact smooth four-di- mensional manifold M such that Diff(M) contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of Diff(N) for compact manifolds N. 1. In [Po10, Sect. 2] was introduced the definition of abstract Jordan group and initiated exploration of the following two problems on automorphism groups and birational self-map groups of algebraic varieties: Problem A. Describe algebraic varieties X for which Aut(X) is Jordan. Problem B. The same with Aut(X) replaced by Bir(X). Recall from [Po10, Def. 2.1] that Jordaness of a group is defined as follows: Definition 1. A group G is called a Jordan group if there exists a positive integer dG, depending on G only, such that every finite subgroup F of G contains a normal abelian subgroup whose index in F is at most dG. Informally Jordaness means that all finite subgroups of G are “almost” abelian in the sense that they are extensions of abelian groups by finite groups taken from a finite list. Since the time of launching the exploration program of Problems A and B a rather extensive information on them has been obtained (see [Po132]), but, for instance, at this writing (October 2013) is still unknown whether there exists X with non-Jordan group Aut(X) (see Question 1 in [Po132]; it is interesting to juxtapose it with Corollary 2 below). 2. In this note we explore the counterpart of Problem A, in which algebraic varieties X are replaced by connected smooth topological manifolds M, and arXiv:1310.6548v1 [math.GR] 24 Oct 2013 Aut(X) is replaced by Diff(M), the diffeomorphism group of M. -
The Theory of Finite Groups: an Introduction (Universitext)
Universitext Editorial Board (North America): S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo This page intentionally left blank Hans Kurzweil Bernd Stellmacher The Theory of Finite Groups An Introduction Hans Kurzweil Bernd Stellmacher Institute of Mathematics Mathematiches Seminar Kiel University of Erlangen-Nuremburg Christian-Albrechts-Universität 1 Bismarckstrasse 1 /2 Ludewig-Meyn Strasse 4 Erlangen 91054 Kiel D-24098 Germany Germany [email protected] [email protected] Editorial Board (North America): S. Axler F.W. Gehring Mathematics Department Mathematics Department San Francisco State University East Hall San Francisco, CA 94132 University of Michigan USA Ann Arbor, MI 48109-1109 [email protected] USA [email protected] K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA [email protected] Mathematics Subject Classification (2000): 20-01, 20DXX Library of Congress Cataloging-in-Publication Data Kurzweil, Hans, 1942– The theory of finite groups: an introduction / Hans Kurzweil, Bernd Stellmacher. p. cm. — (Universitext) Includes bibliographical references and index. ISBN 0-387-40510-0 (alk. paper) 1. Finite groups. I. Stellmacher, B. (Bernd) II. Title. QA177.K87 2004 512´.2—dc21 2003054313 ISBN 0-387-40510-0 Printed on acid-free paper. © 2004 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. -
Kernels of Linear Representations of Lie Groups, Locally Compact
Kernels of Linear Representations of Lie Groups, Locally Compact Groups, and Pro-Lie Groups Markus Stroppel Abstract For a topological group G the intersection KOR(G) of all kernels of ordinary rep- resentations is studied. We show that KOR(G) is contained in the center of G if G is a connected pro-Lie group. The class KOR(C) is determined explicitly if C is the class CONNLIE of connected Lie groups or the class ALMCONNLIE of almost con- nected Lie groups: in both cases, it consists of all compactly generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as KOR(G) for some connected pro-Lie group G. However, the dimension of KOR(G) is bounded by the cardinality of the continuum if G is locally compact and connected. Examples are given to show that KOR(C) becomes complicated if C contains groups with infinitely many connected components. 1 The questions we consider and the answers that we have found In the present paper we study (Hausdorff) topological groups. Ifall else fails, we endow a group with the discrete topology. For any group G one tries, traditionally, to understand the group by means of rep- resentations as groups of matrices. To this end, one studies the continuous homomor- phisms from G to GLnC for suitable positive integers n; so-called ordinary representa- tions. This approach works perfectly for finite groups because any such group has a faithful ordinary representation but we may face difficulties for infinite groups; there do exist groups admitting no ordinary representations apart from the trivial (constant) one. -
Arxiv:1305.5974V1 [Math-Ph]
INTRODUCTION TO SPORADIC GROUPS for physicists Luis J. Boya∗ Departamento de F´ısica Te´orica Universidad de Zaragoza E-50009 Zaragoza, SPAIN MSC: 20D08, 20D05, 11F22 PACS numbers: 02.20.a, 02.20.Bb, 11.24.Yb Key words: Finite simple groups, sporadic groups, the Monster group. Juan SANCHO GUIMERA´ In Memoriam Abstract We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups Zp, and the alternating groups Altn>4. After a quick revision of finite fields Fq, q = pf , with p prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 extra “sporadic” groups, which gather in three interconnected “generations” (with 5+7+8 groups) plus the Pariah groups (6). We point out a couple of physical applications, in- cluding constructing the biggest sporadic group, the “Monster” group, with close to 1054 elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory. ∗[email protected] arXiv:1305.5974v1 [math-ph] 25 May 2013 1 Contents 1 Introduction 3 1.1 Generaldescriptionofthework . 3 1.2 Initialmathematics............................ 7 2 Generalities about groups 14 2.1 Elementarynotions............................ 14 2.2 Theframeworkorbox .......................... 16 2.3 Subgroups................................. 18 2.4 Morphisms ................................ 22 2.5 Extensions................................. 23 2.6 Familiesoffinitegroups ......................... 24 2.7 Abeliangroups .............................. 27 2.8 Symmetricgroup ............................. 28 3 More advanced group theory 30 3.1 Groupsoperationginspaces. 30 3.2 Representations.............................. 32 3.3 Characters.Fourierseries . 35 3.4 Homological algebra and extension theory . 37 3.5 Groupsuptoorder16.......................... -
Construction of the Rudvalis Group of Order 145,926,144,000*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 27, 538-548 (1973) Construction of the Rudvalis Group of Order 145,926,144,000* J. H. CONWAY Department of Pure Mathematics and Mathematical Statistics, Cambridge, England AND D. B. WALES Department of Mathematics, California Institute of Technology, Pasadena, California 91109 Communicated by Marshall Hall, Jr. Received August 4, 1972 Recently, Arunas Rudvalis [l] provided evidence for the existence of a new simple group R of order 145,926,144,000 = 2r4 * 33 * 5a . 7 * 13 . 29. He describes the group as a rank 3 permutation group on 4060 letters, in which the stabilizer of a point is the (nonsimple) Ree group F = V’,(2), which has orbits of sizes 1,1755,2304, corresponding to subgroups of F of orders 20480 and 15600. The first of these is the centralizer of an involution of F, while the second has the known subgroup L = PSLa(25) of P’ (see Ref. [4]) as a subgroup of index 2. Since one of the involutions in R has no fixed point and so just 2030 2-cycles, an argument of Griess and Schur [2] shows that R has a proper double cover 2R. Rudvalis and Frame gave evidence for supposing that this group had a 28-dimensional complex representation not splitting over F. (Feit and Lyons have proved this under further assumptions on R.) We sup- pose that this unitary representation of 2R exists while constructing the group, but then conclude independently of this supposition that the con- struction defines a group 2R whose central quotient is R. -
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. -
Cyclotomy and Difference Families in Elementary Abelian Groups*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF NUMBER THEORY 4, 17-47 (1972) Cyclotomy and Difference Families in Elementary Abelian Groups* RICHARD M. WILSON+ Department of Mathematics, Ohio State University, Columbus, Ohio 43210 Communicated by A. C. Woods Received May 1, 1970 By a (v, k, h)-difference family in an additive abelian group G of order v, we mean a family (4 j i E Z) of subsets of G, each of cardinality k, and such that among the differences (a - b / q b E Z3<; a # b; i E Z) each nonzero g E G occurs X times. The existence of such a difference family implies the existence of a (v, k, X)-BIBD with G as a regular group of automorphisms. The multiplicative structure of finite fields is used here to construct difference families in their additive groups. It is proven that if q is a prime power and q > {ik(k - l)}klk-l), then there exists a (q, k, h)-difference family in the elementary abelian group of order q iff X(q - 1) = 0 (mod k(k - 1)). Some extensions of known con- structions and partially balanced designs are also discussed. 1. BALANCED INCOMPLETE BLOCK DESIGNS An incidence structure is a triple (P, L, 9) where P is a set of points, L is a set of lines, and Y is an incidence relation between P and L which we view as a mapping from P x L into the set of integers (0, I}. -
Finite Simple Groups Which Projectively Embed in an Exceptional Lie Group Are Classified!
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 1, January 1999, Pages 75{93 S 0273-0979(99)00771-5 FINITE SIMPLE GROUPS WHICH PROJECTIVELY EMBED IN AN EXCEPTIONAL LIE GROUP ARE CLASSIFIED! ROBERT L. GRIESS JR. AND A. J. E. RYBA Abstract. Since finite simple groups are the building blocks of finite groups, it is natural to ask about their occurrence “in nature”. In this article, we consider their occurrence in algebraic groups and moreover discuss the general theory of finite subgroups of algebraic groups. 0. Introduction Group character theory classifies embeddings of finite groups into classical groups. No general theory classifies embeddings into the exceptional complex al- gebraic group, i.e., one of G2(C), F4(C), E6(C), E7(C), E8(C). For exceptional groups, special methods seem necessary. Since the early 80s, there have been ef- forts to determine which central extensions of finite simple groups embed in an exceptional group. For short, we call this work a study of projective embeddings of finite simple groups into exceptional groups. Table PE on page 84 contains a summary. The classification program for finite subgroups of complex algebraic groups involves both existence of embeddings and their classification up to conjugacy. We have just classified embeddings of Sz(8) into E8(C) (there are three, up to E8(C)- conjugacy), thus settling the existence question for projective embeddings of finite simple groups into exceptional algebraic groups. The conjugacy part of the program is only partially resolved. The finite subgroups of the smallest simple algebraic group PSL(2; C)(upto conjugacy) constitute the famous list: cyclic, dihedral, Alt4, Sym4, Alt5. -
Moonshines for L2(11) and M12
Moonshines for L2(11) and M12 Suresh Govindarajan Department of Physics Indian Institute of Technology Madras (Work done with my student Sutapa Samanta) Talk at the Workshop on Modular Forms and Black Holes @NISER, Bhubaneswar on January 13. 2017 Plan Introduction Some finite group theory Moonshine BKM Lie superalgebras Introduction Classification of Finite Simple Groups Every finite simple group is isomorphic to one of the following groups: (Source: Wikipedia) I A cyclic group with prime order; I An alternating group of degree at least 5; I A simple group of Lie type, including both I the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field; I the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lie type). I The 26 sporadic simple groups. The classification was completed in 2004 when Aschbacher and Smith filled the last gap (`the quasi-thin case') in the proof. Fun Reading: Symmetry and the Monster by Mark Ronan The sporadic simple groups I the Mathieu groups: M11, M12, M22, M23, M24; (found in 1861) I the Janko groups: J1, J2, J3, J4; (others 1965-1980) I the Conway groups; Co1, Co2, Co3; 0 I the Fischer groups; Fi22. Fi23, Fi24; I the Higman-Sims group; HS I the McLaughlin group: McL I the Held group: He; I the Rudvalis group Ru; I the Suzuki sporadic group: Suz; 0 I the O'Nan group: O N; I Harada-Norton group: HN; I the Lyons group: Ly; I the Thompson group: Th; I the baby Monster group: B and Sources: Wikipedia and Mark Ronan I the Fischer-Griess Monster group: M Monstrous Moonshine Conjectures I The j-function has the followed q-series: (q = exp(2πiτ)) j(τ)−744 = q−1+[196883+1] q+[21296876+196883+1] q2+··· I McKay observed that 196883 and 21296876 are the dimensions of the two smallest irreps of the Monster group.