An Alternative Existence Proof of the Geometry of Ivanov–Shpectorov for O'nan's Sporadic Group
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The Book of Abstracts
1st Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society Victoria University of Wellington Wellington, New Zealand December 12{15, 2007 American Mathematical Society New Zealand Mathematical Society Contents Timetables viii Plenary Addresses . viii Special Sessions ............................. ix Computability Theory . ix Dynamical Systems and Ergodic Theory . x Dynamics and Control of Systems: Theory and Applications to Biomedicine . xi Geometric Numerical Integration . xiii Group Theory, Actions, and Computation . xiv History and Philosophy of Mathematics . xv Hopf Algebras and Quantum Groups . xvi Infinite Dimensional Groups and Their Actions . xvii Integrability of Continuous and Discrete Evolution Systems . xvii Matroids, Graphs, and Complexity . xviii New Trends in Spectral Analysis and PDE . xix Quantum Topology . xx Special Functions and Orthogonal Polynomials . xx University Mathematics Education . xxii Water-Wave Scattering, Focusing on Wave-Ice Interactions . xxiii General Contributions . xxiv Plenary Addresses 1 Marston Conder . 1 Rod Downey . 1 Michael Freedman . 1 Bruce Kleiner . 2 Gaven Martin . 2 Assaf Naor . 3 Theodore A Slaman . 3 Matt Visser . 4 Computability Theory 5 George Barmpalias . 5 Paul Brodhead . 5 Cristian S Calude . 5 Douglas Cenzer . 6 Chi Tat Chong . 6 Barbara F Csima . 6 QiFeng ................................... 6 Johanna Franklin . 7 Noam Greenberg . 7 Denis R Hirschfeldt . 7 Carl G Jockusch Jr . 8 Bakhadyr Khoussainov . 8 Bj¨ornKjos-Hanssen . 8 Antonio Montalban . 9 Ng, Keng Meng . 9 Andre Nies . 9 i Jan Reimann . 10 Ludwig Staiger . 10 Frank Stephan . 10 Hugh Woodin . 11 Guohua Wu . 11 Dynamical Systems and Ergodic Theory 12 Boris Baeumer . 12 Mathias Beiglb¨ock . 12 Arno Berger . 12 Keith Burns . 13 Dmitry Dolgopyat . 13 Anthony Dooley . -
Bogomolov F., Tschinkel Yu. (Eds.) Geometric Methods in Algebra And
Progress in Mathematics Volume 235 Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein Geometric Methods in Algebra and Number Theory Fedor Bogomolov Yuri Tschinkel Editors Birkhauser¨ Boston • Basel • Berlin Fedor Bogomolov Yuri Tschinkel New York University Princeton University Department of Mathematics Department of Mathematics Courant Institute of Mathematical Sciences Princeton, NJ 08544 New York, NY 10012 U.S.A. U.S.A. AMS Subject Classifications: 11G18, 11G35, 11G50, 11F85, 14G05, 14G20, 14G35, 14G40, 14L30, 14M15, 14M17, 20G05, 20G35 Library of Congress Cataloging-in-Publication Data Geometric methods in algebra and number theory / Fedor Bogomolov, Yuri Tschinkel, editors. p. cm. – (Progress in mathematics ; v. 235) Includes bibliographical references. ISBN 0-8176-4349-4 (acid-free paper) 1. Algebra. 2. Geometry, Algebraic. 3. Number theory. I. Bogomolov, Fedor, 1946- II. Tschinkel, Yuri. III. Progress in mathematics (Boston, Mass.); v. 235. QA155.G47 2004 512–dc22 2004059470 ISBN 0-8176-4349-4 Printed on acid-free paper. c 2005 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. -
Title: Algebraic Group Representations, and Related Topics a Lecture by Len Scott, Mcconnell/Bernard Professor of Mathemtics, the University of Virginia
Title: Algebraic group representations, and related topics a lecture by Len Scott, McConnell/Bernard Professor of Mathemtics, The University of Virginia. Abstract: This lecture will survey the theory of algebraic group representations in positive characteristic, with some attention to its historical development, and its relationship to the theory of finite group representations. Other topics of a Lie-theoretic nature will also be discussed in this context, including at least brief mention of characteristic 0 infinite dimensional Lie algebra representations in both the classical and affine cases, quantum groups, perverse sheaves, and rings of differential operators. Much of the focus will be on irreducible representations, but some attention will be given to other classes of indecomposable representations, and there will be some discussion of homological issues, as time permits. CHAPTER VI Linear Algebraic Groups in the 20th Century The interest in linear algebraic groups was revived in the 1940s by C. Chevalley and E. Kolchin. The most salient features of their contributions are outlined in Chapter VII and VIII. Even though they are put there to suit the broader context, I shall as a rule refer to those chapters, rather than repeat their contents. Some of it will be recalled, however, mainly to round out a narrative which will also take into account, more than there, the work of other authors. §1. Linear algebraic groups in characteristic zero. Replicas 1.1. As we saw in Chapter V, §4, Ludwig Maurer thoroughly analyzed the properties of the Lie algebra of a complex linear algebraic group. This was Cheval ey's starting point. -
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. -
Lifting a 5-Dimensional Representation of $ M {11} $ to a Complex Unitary Representation of a Certain Amalgam
Lifting a 5-dimensional representation of M11 to a complex unitary representation of a certain amalgam Geoffrey R. Robinson July 16, 2018 Abstract We lift the 5-dimensional representation of M11 in characteristic 3 to a unitary complex representation of the amalgam GL(2, 3) ∗D8 S4. 1 The representation It is well known that the Mathieu group M11, the smallest sporadic simple group, has a 5-dimensional (absolutely) irreducible representation over GF(3) (in fact, there are two mutually dual such representations). It is clear that this does not lift to a complex representation, as M11 has no faithful complex character of degree less than 10. However, M11 is a homomorphic image of the amalgam G = GL(2, 3) D8 S4, ∗ and it turns out that if we consider the 5-dimension representation of M11 as a representation of G, then we may lift that representation of G to a complex representation. We aim to do that in such a way that the lifted representation Z 1 is unitary, and we realise it over [ √ 2 ], so that the complex representation admits reduction (mod p) for each odd− prime. These requirements are stringent enough to allow us explicitly exhibit representing matrices. It turns out that reduction (mod p) for any odd prime p other than 3 yields either a 5-dimensional arXiv:1502.00520v3 [math.RT] 8 Feb 2015 special linear group or a 5-dimensional special unitary group, so it is only the behaviour at the prime 3 which is exceptional. We are unsure at present whether the 5-dimensional complex representation of G is faithful (though it does have free kernel), so we will denote the image of Z 1 G in SU(5, [ √ 2 ]) by L, and denote the image of L under reduction (mod p) − by Lp. -
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. -
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. -
Finite Groups with a Standard Component of Type Jan Ko-Ree
JOURNAL OF ALGEBRA 36, 416-426 (1975) Finite Groups with a Standard Component of Type Jan ko-Ree LARRY FIN-STEIN Wayne State University, Detroit, Michigan 48202 Communicated by Marshall Hall, Jr. Received May 20, 1974 1. INTRODUCTION As in [2], K is tightly embedded in G if K has even order while K n Kg has odd order for g E G - N,(K). A quasisimple group A is standard in G if K = C,(A) is tightly embedded in G, No(K) = N,(A) and [A, Ag] # 1 for g E: G. The main result of this paper is the following. THEOREM A. Let G be a finite group with O(G) = 1 and suppose G contains a standard component A isomorphic to a group of type Janko-Ree. Let X = (AG> and assume X # A. Then either (i) Xr O-S, the O’N an-Sims simple group and G s Aut(O-S); OY (ii) X E G2(32”+1), n > 1, and G is isomorphic to a subgroup of Aut(G2(32n+1)); OY (iii) X z A x A. COROLLARY. Let G be a finite group with Z*(G) = 1 azd suppose for some involution x of G, C,(z) = (z} X A, where A is of type Janko-Ree. Then G is isomorphic to one of the following groups: 0) A 1 ZZ , (ii) Aut(O-S), oydey’p G2(3z”+1)<+, n > 1, where Q induces an outer automorphism of The Corollary contains the statement of our original result. However, recent work of Aschbacher [2] suggests that the hypotheses of Theorem A are more suitable in terms of general classification theory. -
Experimental Mathematics
A Complete Bibliography of Publications in Experimental Mathematics Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 30 March 2021 Version 1.28 Title word cross-reference (((X2 − P )2 − Q)2 − R)2 − S2 [Bre08a]. (−2; 3;n) [GK12]. (1; 2) [EF18]. (2; 3; 6) [Sch06]. (4k + 2) [AJK08]. {−1; 0; 1g [SS06]. 0 [Cum04, Har15]. 1 [BR18, BM00, CF95, Cum04, Fin92]. 1=2 [Car03]. 1/π [Gui06a, Gui06b]. 1/π2 [AG12, Gui06b]. 100 [BCVZ02]. 10651 [MR95]. 1019 [AD07]. 1027 [RT16]. 11 [BLP09, BGKT05, Lep93, LM19]. 1132 [KP08]. 12 [GP94, LM19]. 128 [JM14, Elk01]. 15 [BGKT05, DL92, KO95]. 16 [EMRSE14,´ MTC14]. 19 [CHLM97]. 191 [HPP08]. 1 <n<10 [OM08]. 1: 4 [Kra94]. 2 [AT00, AMM08, BLP09, CD09, Elk01, EL14, KdC05, KV17, Kat10b, Man18, Nic18, RW11, Van17]. 23 [DGKMY15]. 24 [CP03]. 2 × 2 [CILO19, LR17]. 3 [ABL17b, AT00, Ban06, BFG+13, BL00, Bur04, But05, CLLM07, CLT06, CGHN00, Der15, DBG19, EM02, FMP04, Fuk11, GHL19, HS15, HIK+16, Kat10b, KY17, LR07, Sha19, Tan09, Ume09, Web97]. 32 [BN97, CH08a]. 3630 [BCVZ02]. 3n + 1 [LSW99]. 3x + 1 [AL95, BM98, Nic18]. 3x + d [BM98]. 4 [Con06, JK01, RT00, SZ04]. 47 [Tak14]. 5 [De 15a, HS07, Kat10b]. 6 [ALT+12, FP92, FP93, FPDH98, HP08]. 6; 6 [Nil03]. 7 [ALT+12, BGKT05]. 1 2 71 [Tak14]. 8 [ALT+12, BN97, Bur07, FJ96]. -
International Conference Mathematics Days in Sofia
Institute of Mathematics and Informatics Bulgarian Academy of Sciences International Conference Mathematics Days in Sofia July 7–10, 2014, Sofia, Bulgaria Abstracts Sofia, 2014 Programme Committee Mini-symposia Organizers Algebra, Logic, and Combinatorics Advanced Analytical and Numerical Techniques for Applications Vesselin Drensky, Bulgaria – Chair Ludmil Katzarkov, USA Raytcho Lazarov, USA Analysis, Geometry, and Topology Algebraic Methods Oleg Mushkarov, Bulgaria in Quantum Field Theory Luchezar Stoyanov, Australia Vladimir Dobrev, Bulgaria Stanimir Troyanski, Spain Differential Equations Approximation Theory and Special and Mathematical Physics Functions – 2nd series Emil Horozov, Bulgaria Oktay Duman, Turkey Peter Popivanov, Bulgaria Esra Erku¸s-Duman, Turkey Georgi Popov, France Geometry Days in Sofia Mathematical Modeling Vestislav Apostolov, Canada Asen Donchev, USA Johann Davidov, Bulgaria Raytcho Lazarov, USA Gueo Grantcharov, USA Blagovest Sendov, Bulgaria Oleg Mushkarov, Bulgaria Nickolay Yanev, Bulgaria Velichka Milousheva, Bulgaria Mathematical Aspects of Computer Science Transform Methods and Special Functions – 7th Avram Eskenazi, Bulgaria Peter Stanchev, USA Virginia Kiryakova, Bulgaria Organising Committee Variational Analysis Julian Revalski, Bulgaria – Chair Asen Donchev, USA Krassimira Ivanova, Bulgaria Alexander Ioffe, Israel Neli Dimitrova, Bulgaria Mikhail Krastanov, Bulgaria Supported by AMERICAN FOUNDATION FOR BULGARIA NOVA TRADE Ltd. Copyright © 2014 Institute of Mathematics and Informatics of BAS, Sofia Content Preface -
The Character Table of a Sharply 5-Transitive Subgroup of the Alternating Group of Degree 12
International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 www.theoryofgroups.ir Vol. x No. x (201x), pp. xx-xx. www.ui.ac.ir c 201x University of Isfahan THE CHARACTER TABLE OF A SHARPLY 5-TRANSITIVE SUBGROUP OF THE ALTERNATING GROUP OF DEGREE 12 NICK GILL AND SAM HUGHES Communicated by Abstract. We calculate the character table of a sharply 5-transitive subgroup of Alt(12), and of a sharply 4-transitive subgroup of Alt(11). Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations. 1. Introduction Let Alt(n) denote the alternating group on n points. In this paper we will be interested in subgroups of Alt(n) that are sharply k-transitive, for some integer k ≥ 4. Recall that a permutation group G acting on a set Ω is sharply k-transitive if, first, G acts transitively on the set of k-tuples of distinct elements of Ω and, second, the (point-wise) stabilizer of any k-tuple of distinct elements of Ω is trivial. Notice that \1-transitivity" is the same as \transitivity". In this paper we prove the following theorem. Theorem 1.1. (1) If G is a sharply 5-transitive subgroup of Alt(12), then its character table is given by Table 1. (2) If G is a sharply 4-transitive subgroup of Alt(11), then its character table is given by Table 2. -
The Topological Index and Automorphism Group of 1,3,5
Available at http://pvamu.edu/aam Applications and Applied Appl. Appl. Math. Mathematics: ISSN: 1932-9466 An International Journal (AAM) Vol. 11, Issue 2 (December 2016), pp. 930 - 942 Study on the Q-Conjugacy Relations for the Janko Groups Ali Moghani Department of Computer Science William Paterson University Wayne, NJ 07470 USA E-mail: [email protected] Received March 5, 2016; Accepted July 22, 2016 ABSTRACT In this paper, we consider all the Janko sporadic groups J1, J2, J3 and J4 (with orders 175560, 604800, 50232960 and 86775571046077562880, respectively) with a new concept called the markaracter- and Q-conjugacy character tables, which enables us to discuss marks and characters for a finite group on a common basis of Q-conjugacy relationships between their cyclic subgroups. Then by using GAP (Groups, Algorithms and Programming) package we calculate all their dominant classes enabling us to find all possible Q-conjugacy characters for these sporadic groups. Finally, we prove in a main theorem that all twenty six simple sporadic groups are unmatured. Keywords: Finite group; Sporadic, Janko; Conjugacy class; Character, Q-conjugacy; matured MSC 2010: 20D99, 20C15 1. Introduction In recent years, group theory has drawn wide attention of researchers in mathematics, physics and chemistry, see Fujita (1998). Many problems of the computational group theory have been researched, such as the classification, the symmetry, the topological cycle index, etc. They include not only the diverse properties of finite groups, but also their wide-ranging connections with many applied sciences, such as Nanoscience, Chemical Physics and Quantum Chemistry. For instance, see Darafsheh et al.