Transposition Algebras
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On Some Generation Methods of Finite Simple Groups
Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography On Some Generation Methods of Finite Simple Groups Ayoub B. M. Basheer Department of Mathematical Sciences, North-West University (Mafikeng), P Bag X2046, Mmabatho 2735, South Africa Groups St Andrews 2017 in Birmingham, School of Mathematics, University of Birmingham, United Kingdom 11th of August 2017 Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Abstract In this talk we consider some methods of generating finite simple groups with the focus on ranks of classes, (p; q; r)-generation and spread (exact) of finite simple groups. We show some examples of results that were established by the author and his supervisor, Professor J. Moori on generations of some finite simple groups. Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Introduction Generation of finite groups by suitable subsets is of great interest and has many applications to groups and their representations. For example, Di Martino and et al. [39] established a useful connection between generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. Their motivation was to study irreducible projective representations of the sporadic simple groups. In view of applications, it is often important to exhibit generating pairs of some special kind, such as generators carrying a geometric meaning, generators of some prescribed order, generators that offer an economical presentation of the group. -
The Centre of a Block
THE CENTRE OF A BLOCK A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2016 Inga Schwabrow School of Mathematics 2 Contents Abstract 5 Declaration 7 Copyright Statement 9 Acknowledgements 11 Introduction 13 1 Modular Representation Theory 17 1.1 Basic notation and setup . 17 1.2 Conjugacy class sums, characters and Burnside's formula . 18 1.3 Blocks . 22 1.4 Defect and defect groups . 25 1.5 Jacobson radical and Loewy length . 28 1.6 The Reynolds ideal . 35 2 Equivalences of blocks 37 2.1 Brauer correspondence . 37 2.2 Morita equivalence . 38 2.3 Derived equivalence . 38 2.4 Stable equivalence of Morita type . 40 2.5 Properties of blocks with TI defect groups . 41 2.6 On using the centre to show no derived equivalence exists . 43 3 Blocks with trivial intersection defect groups 44 3 3.1 The Mathieu group M11 with p =3.................... 45 3.2 The McLaughlin group McL, and Aut(McL), with p = 5 . 49 3.3 The Janko group J4 with p =11...................... 52 3.4 The projective special unitary groups . 54 3.5 A question of Rickard . 60 3.6 On the existence of perfect isometries in blocks with TI defect groups . 62 4 On the Loewy length of the Suzuki groups and the small Ree groups in defining characteristic 68 4.1 Relating structure constants . 68 4.2 Suzuki Groups . 70 4.3 The Ree groups . 76 5 More sporadic simple groups 108 5.1 Mathieu groups . -
CHARACTERS of Π1-DEGREE with CYCLOTOMIC FIELDS of VALUES 1. Introduction in 2006, G. Navarro and P.H. Tiep [NT06] Confirmed
CHARACTERS OF π1-DEGREE WITH CYCLOTOMIC FIELDS OF VALUES EUGENIO GIANNELLI, NGUYEN NGOC HUNG, A. A. SCHAEFFER FRY, AND CAROLINA VALLEJO Abstract. We characterize finite groups that possess a nontrivial irreducible character of tp, qu1-degree with values in Qpe2πi{pq or Qpe2πi{qq, where p and q are primes. This extends previous work of Navarro-Tiep and of Giannelli-Schaeffer Fry-Vallejo. Along the way we completely describe the alternating groups possessing a nontrivial irreducible rational-valued character of tp, qu1-degree. A similar classification is obtained for solvable groups, when p “ 2. 1. Introduction In 2006, G. Navarro and P.H. Tiep [NT06] confirmed a conjecture of R. Gow predicting that every group of even order has a nontrivial rational-valued irreducible character of odd degree. Later, in [NT08], they generalized their result by proving that every finite group of order divisible by a prime q admits a nontrivial irreducible character of q1-degree with values in Qpe2iπ{qq, the cyclotomic field extending the rational numbers by a primitive q-th root of unity. In [GSV19], the first, third, and fourth-named authors have recently shown that for any set π consisting of at most two primes, every nontrivial group has a nontrivial character of π1-degree (that is, a character of p1-degree, for all primes p P π). How to extend these results, if possible, is the main topic under consideration in this article. In Theorem A, we show that every finite group possesses a nontrivial irreducible character of t2, qu1-degree with values in Qpe2iπ{qq, an unexpected result that generalizes both [NT06] and [NT08] in the fashion of [GSV19]. -
The 5-Modular Representations of the Tits Simple Group in the Principal Block
mathematics of computation volume 57,number 195 july 1991,pages 369-386 THE 5-MODULARREPRESENTATIONS OF THE TITS SIMPLE GROUP IN THE PRINCIPAL BLOCK holger w. gollan Abstract. In this paper we show how to construct the 5-modular absolutely irreducible representations of the Tits simple group in the principal block, which is the only block of positive defect. Starting with the smallest nontrivial ones, all the others except one pair are obtained as constituents of tensor products of dimension at most 729. The last two we get from a permutation representation of degree 1600. We give an exact description of the construction of the first one of degree 26 by extending its restrictions to several subgroups, a method first used in the existence proof of the Janko group J4 . Using the explicit matrices obtained from the above constructions, we work out the Green correspondents and sources of all the representations and state their socle series. 0. Introduction One aim of modular representation theory is the construction of the p- modular absolutely irreducible representations of finite groups G, where p is a prime dividing the group order \G\. Having done this, invariants of these representations like vertices, Green correspondents, and sources may be com- puted to get more information about them. This paper deals with the 5-modular representations of the Tits simple group -E4(2)' in the principal block. This is in some sense the only interesting block for this problem in E4(2)', since all the other blocks are blocks of defect 0. The following tables show some small permutation representations and tensor products of E4(2)' written as sums of their 5-modular constituents. -
Centres of Blocks of Finite Groups with Trivial Intersection Sylow $ P
Centres of blocks of finite groups with trivial intersection Sylow p-subgroups Inga Schwabrow April 14, 2018 FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany E-mail: [email protected] Abstract For finite groups G with non-abelian, trivial intersection Sylow p-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of G and the centre of the Brauer correspondent. This was already known for the Suzuki groups; the result will be generalised to cover more groups with trivial intersection Sylow p-subgroups. Keywords: blocks, trivial intersection defect groups, stable equivalence of Morita type AMS classification: 20C05, 20C15 1 Introduction Let p be a fixed prime number. Let G be a finite group whose order is divisible by p, and P a Sylow p- subgroup of G. Throughout, (K, O, k) is a p-modular system; in particular, k is an algebraically closed field of characteristic p and O is a complete valuation ring whose residue field k has characteristic p. Local representation theory studies the connection between kG and kNG(P ); Brou´e’s abelian de- fect group conjecture (ADGC) predicts the existence of a derived equivalence between the principal blocks B0(kG) and b0(kNG(P )) if the Sylow p-subgroups of G are abelian. We investigate the ques- tion of whether a derived equivalence can exist when the Sylow p-subgroups are non-abelian, trivial intersection subgroups of G. -
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. -
Preface Issue 2-2013
Jahresber Dtsch Math-Ver (2013) 115:61–62 DOI 10.1365/s13291-013-0068-0 PREFACE Preface Issue 2-2013 Hans-Christoph Grunau © Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2013 “Curvature driven interface evolution” is one of the major current themes in the nu- merics resp. the numerical analysis resp. the analytical theory of partial differential equations. It is particularly attractive due to numerous physical applications on the one hand and due to close relations to differential geometry on the other hand. Harald Garcke gives a survey on the most important topics in this research area. Among these are models for crystal growth of snow flakes, melting ice in water or grain coarsening (aging) in two-phase mixtures: Stefan problem, Mullins-Sekerka-problem and vari- ants. Moreover, the mean curvature and surface diffusion flow are covered as well as phase field models like the Allen-Cahn and Cahn-Hilliard equation. In the latter sharp classical interfaces are replaced by interfacial regions. All of these models are introduced, their physical applicability as well as their main properties are explained and they are illustrated by means of numerical simulations. The present survey will certainly prove to be a very helpful reference work for anybody working on resp. interested in these kinds of models. The integral group ring ZG of a group G consists of elements of the form ∈ Z g∈G zgg, where only finitely many of the coefficients zg may be different from 0. These elements form a free Z-module with G as a basis, while the multi- plication in ZG is the Z-linear extension of that in G. -
The Logbook of Pointed Hopf Algebras Over the Sporadic Groups
THE LOGBOOK OF POINTED HOPF ALGEBRAS OVER THE SPORADIC SIMPLE GROUPS N. ANDRUSKIEWITSCH, F. FANTINO, M. GRANA˜ AND L. VENDRAMIN Contents Introduction 1 1. Algorithms 2 1.1. Algorithms for type D 2 1.2. Structure constants 4 1.3. Bases for permutation groups 4 1.4. An algorithm for involutions 4 2. Some auxiliary groups 4 2.1. The groups A9, A11, A12, S12 5 2.2. The group L5(2) 5 2.3. The group O7(3) 5 + 2.4. The group O8 (2) 5 − 2.5. The group O10(2) 6 2.6. The exceptional group G2(4) 6 2.7. The exceptional groups G2(3) and G2(5) 7 2.8. The symplectic group S6(2) 7 2.9. The symplectic group S8(2) 7 2.10. TheautomorphismgroupoftheTitsgroup 7 2.11. Direct products 8 3. Proof of Theorem II 8 3.1. The Lyons group Ly 8 3.2. The Thompson group Th 10 3.3. The Janko group J4 11 3.4. The Fischer group Fi23 12 3.5. The Conway group Co1 13 3.6. The Harada-Norton group HN 14 ′ 3.7. The Fischer group Fi24 14 3.8. The Baby Monster group B 15 arXiv:1001.1113v2 [math.QA] 25 Oct 2010 3.9. The Monster group M 17 Appendix.Realandquasi-realconjugacyclasses 18 References 21 Introduction In these notes, we give details on the proofs performed with GAP of the theorems of our paper [AFGV2]. Here we discuss the algorithms implemented for studying Date: October 31, 2018. 2000 Mathematics Subject Classification. 16W30; 17B37. -
Pi Formulas, Ramanujan, and the Baby Monster Group By
Pi Formulas, Ramanujan, and the Baby Monster Group By Titus Piezas III Keywords: Pi formulas, class polynomials, j-function, modular functions, Ramanujan, finite groups. Contents I. Introduction II. Pi Formulas A. The j-function and Hilbert Class Polynomials B. Weber Class Polynomials C. Ramanujan Class Polynomials III. Baby Monster Group IV. Conclusion I. Introduction In 1914, Ramanujan wrote a fascinating article in the Quarterly Journal of Pure and Applied Mathematics. The title was “Modular equations and approximations to p” and he discusses certain methods to derive exact and approximate evaluations. In fact, in his Notebooks, he gave several formulas, seventeen in all, for p (or to be more precise, 1/p). For a complete list, see http://mathworld.wolfram.com/PiFormulas.html. He gave little explanation on how he came up with them, other than saying that there were “corresponding theories”. One of the more famous formulas is given by, 1/(pÖ8) = 1/992 S r (26390n+1103)/3964n where r = (4n)!/(n!4) and the sum S (from this point of the article onwards) is to go from n = 0 to ¥. How did Ramanujan find such beautiful formulas? We can only speculate at the heuristics he used to find them, since they were only rigorously proven to be true in 1987 by the Borwein brothers. However, one thing we do know: Ramanujan wrote down in his Notebooks, among the others, the approximation1, epÖ58 » 24591257751.9999998222… which can also be stated as, epÖ58 » 3964 – 104 The appearance of 3964 in both the exact formula and the approximation is no coincidence. -
Words for Maximal Subgroups of Fi '
Open Chem., 2019; 17: 1491–1500 Research Article Faisal Yasin, Adeel Farooq, Chahn Yong Jung* Words for maximal Subgroups of Fi24‘ https://doi.org/10.1515/chem-2019-0156 received October 16, 2018; accepted December 1, 2019. energy levels, and even bond order to name a few can be found, all without rigorous calculations [2]. The fact that Abstract: Group Theory is the mathematical application so many important physical aspects can be derived from of symmetry to an object to obtain knowledge of its symmetry is a very profound statement and this is what physical properties. The symmetry of a molecule provides makes group theory so powerful [3,4]. us with the various information, such as - orbitals energy The allocated point groups would then be able to levels, orbitals symmetries, type of transitions than can be utilized to decide physical properties, (for example, occur between energy levels, even bond order, all that concoction extremity and chirality), spectroscopic without rigorous calculations. The fact that so many properties (especially valuable for Raman spectroscopy, important physical aspects can be derived from symmetry infrared spectroscopy, round dichroism spectroscopy, is a very profound statement and this is what makes mangnatic dichroism spectroscopy, UV/Vis spectroscopy, group theory so powerful. In group theory, a finite group and fluorescence spectroscopy), and to build sub-atomic is a mathematical group with a finite number of elements. orbitals. Sub-atomic symmetry is in charge of numerous A group is a set of elements together with an operation physical and spectroscopic properties of compounds and which associates, to each ordered pair of elements, an gives important data about how chemical reaction happen. -
On the Action of the Sporadic Simple Baby Monster Group on Its Conjugacy Class 2B
London Mathematical Society ISSN 1461–1570 ON THE ACTION OF THE SPORADIC SIMPLE BABY MONSTER GROUP ON ITS CONJUGACY CLASS 2B JURGEN¨ MULLER¨ Abstract We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity- free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element 1+22 is a maximal subgroup of shape 2 .Co2.Thisisoneofthe first applications of a new general computational technique to enumerate big orbits. 1. Introduction The aim of the present work is to determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the Baby Monster group B, the second largest of the sporadic simple groups, on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup 1+22 of shape 2 .Co2. The final result is given in Table 4. In general, the endomorphism ring of a permutation module reflects aspects of the representation theory of the underlying group. Its character table in particular encodes information about the spectral properties of the orbital graphs associated with the permutation action, such as distance-transitivity or distance-regularity, see [14, 5], or the Ramanujan property, see [7]. Here multiplicity-free actions, that is those whose associated endomorphism ring is commutative, have been of particular interest; for example a distance-transitive graph necessarily is an orbital graph associated with a multiplicity-free action. The multiplicity-free permutation actions of the sporadic simple groups and the related almost quasi-simple groups have been classified in [3, 16, 2], and the as- sociated character tables including the one presented here have been collected in [4, 20]. -
Verification of the Ordinary Character Table of the Baby Monster 3
VERIFICATION OF THE ORDINARY CHARACTER TABLE OF THE BABY MONSTER THOMAS BREUER, KAY MAGAARD, AND ROBERT A. WILSON Abstract. We prove the correctness of the character table of the sporadic simple Baby Monster group that is shown in the ATLAS of Finite Groups. MSC: 20C15,20C40,20D08 In memory of our friend and colleague Kay Magaard, who sadly passed away during the preparation of this paper. 1. Introduction Jean-Pierre Serre has raised the question of verification of the ordinary character tables that are shown in the ATLAS of Finite Groups [6]. This question was partially answered in the paper [5], the remaining open cases being the largest two sporadic simple groups, the Baby Monster group B and the Monster Group M, and the double cover 2.B of B. The current paper describes a verification of the character table of B. The computations shown in [3] then imply that also the ATLAS character table of 2.B is correct. As in [5], one of our aims is to provide the necessary data in a way that makes it easy to reproduce our computations. The ATLAS character table of the Baby Monster derives from the original cal- culation of the conjugacy classes and rational character table by David Hunt, de- scribed very briefly in [9]. The irrationalities were calculated by the CAS team in Aachen [11]. 2. Strategy We begin with a preliminary section, Section 3, whose aim is to prove that certain specified matrices do indeed generate copies of the Baby Monster. These arXiv:1902.07758v2 [math.RT] 20 May 2019 matrices can then be used in the main computation.