Classifying Spaces of Sporadic Groups
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On the Classification of Finite Simple Groups
On the Classification of Finite Simple Groups Niclas Bernhoff Division for Engineering Sciences, Physics and Mathemathics Karlstad University April6,2004 Abstract This paper is a small note on the classification of finite simple groups for the course ”Symmetries, Groups and Algebras” given at the Depart- ment of Physics at Karlstad University in the Spring 2004. 1Introduction In 1892 Otto Hölder began an article in Mathematische Annalen with the sen- tence ”It would be of the greatest interest if it were possible to give an overview of the entire collection of finite simple groups” (translation from German). This can be seen as the starting point of the classification of simple finite groups. In 1980 the following classification theorem could be claimed. Theorem 1 (The Classification theorem) Let G be a finite simple group. Then G is either (a) a cyclic group of prime order; (b) an alternating group of degree n 5; (c) a finite simple group of Lie type;≥ or (d) one of 26 sporadic finite simple groups. However, the work comprises in total about 10000-15000 pages in around 500 journal articles by some 100 authors. This opens for the question of possible mistakes. In fact, around 1989 Aschbacher noticed that an 800-page manuscript on quasithin groups by Mason was incomplete in various ways; especially it lacked a treatment of certain ”small” cases. Together with S. Smith, Aschbacher is still working to finish this ”last” part of the proof. Solomon predicted this to be finished in 2001-2002, but still until today the paper is not completely finished (see the homepage of S. -
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. -
Finite Groups, Designs and Codes
Abstract Introduction Terminology and notation Group Actions and Permutation Characters Method 1 References Finite Groups, Designs and Codes J Moori School of Mathematical Sciences, University of KwaZulu-Natal Pietermaritzburg 3209, South Africa ASI, Opatija, 31 May –11 June 2010 J Moori, ASI 2010, Opatija, Croatia Groups, Designs and Codes Abstract Introduction Terminology and notation Group Actions and Permutation Characters Method 1 References Finite Groups, Designs and Codes J Moori School of Mathematical Sciences, University of KwaZulu-Natal Pietermaritzburg 3209, South Africa ASI, Opatija, 31 May –11 June 2010 J Moori, ASI 2010, Opatija, Croatia Groups, Designs and Codes Abstract Introduction Terminology and notation Group Actions and Permutation Characters Method 1 References Outline 1 Abstract 2 Introduction 3 Terminology and notation 4 Group Actions and Permutation Characters Permutation and Matrix Representations Permutation Characters 5 Method 1 Janko groups J1 and J2 Conway group Co2 6 References J Moori, ASI 2010, Opatija, Croatia Groups, Designs and Codes Abstract Introduction Terminology and notation Group Actions and Permutation Characters Method 1 References Abstract Abstract We will discuss two methods for constructing codes and designs from finite groups (mostly simple finite groups). This is a survey of the collaborative work by the author with J D Key and B Rorigues. In this talk (Talk 1) we first discuss background material and results required from finite groups, permutation groups and representation theory. Then we aim to describe our first method of constructing codes and designs from finite groups. J Moori, ASI 2010, Opatija, Croatia Groups, Designs and Codes Abstract Introduction Terminology and notation Group Actions and Permutation Characters Method 1 References Abstract Abstract We will discuss two methods for constructing codes and designs from finite groups (mostly simple finite groups). -
Verification of Dade's Conjecture for Janko Group J3
JOURNAL OF ALGEBRA 187, 579]619Ž. 1997 ARTICLE NO. JA966836 Verification of Dade's Conjecture for Janko Group J3 Sonja KotlicaU Department of Mathematics, Uni¨ersity of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801 View metadata, citation and similar papersCommunicated at core.ac.uk by Walter Feit brought to you by CORE provided by Elsevier - Publisher Connector Received July 5, 1996 Inwx 3 Dade made a conjecture expressing the number kBŽ.,d of characters of a given defect d in a given p-block B of a finite group G in terms of the corresponding numbers kbŽ.,d for blocks b of certain p-local subgroups of G. Several different forms of this conjecture are given inwx 5 . Dade claims that the most complicated form of this conjecture, called the ``Inductive Conjecture 5.8'' inwx 5 , will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive conjecture for all covering groups of the third Janko group J3 Žin the notation of the Atlaswx 1. This is one step in the inductive proof of the conjecture for all finite groups. Certain properties of J33simplify our task. The Schur Multiplier of J is cyclic of order 3Ž seewx 1, p. 82. Hence, there are just two covering groups of J33, namely J itself and a central extension 3 ? J33of J by a cyclic group Z of order 3. We treat these two covering groups separately. The outer automorphism group OutŽ.J33of J is cyclic of order 2 Ž seewx 1, p. -
Janko's Sporadic Simple Groups
Janko’s Sporadic Simple Groups: a bit of history Algebra, Geometry and Computation CARMA, Workshop on Mathematics and Computation Terry Gagen and Don Taylor The University of Sydney 20 June 2015 Fifty years ago: the discovery In January 1965, a surprising announcement was communicated to the international mathematical community. Zvonimir Janko, working as a Research Fellow at the Institute of Advanced Study within the Australian National University had constructed a new sporadic simple group. Before 1965 only five sporadic simple groups were known. They had been discovered almost exactly one hundred years prior (1861 and 1873) by Émile Mathieu but the proof of their simplicity was only obtained in 1900 by G. A. Miller. Finite simple groups: earliest examples É The cyclic groups Zp of prime order and the alternating groups Alt(n) of even permutations of n 5 items were the earliest simple groups to be studied (Gauss,≥ Euler, Abel, etc.) É Evariste Galois knew about PSL(2,p) and wrote about them in his letter to Chevalier in 1832 on the night before the duel. É Camille Jordan (Traité des substitutions et des équations algébriques,1870) wrote about linear groups defined over finite fields of prime order and determined their composition factors. The ‘groupes abéliens’ of Jordan are now called symplectic groups and his ‘groupes hypoabéliens’ are orthogonal groups in characteristic 2. É Émile Mathieu introduced the five groups M11, M12, M22, M23 and M24 in 1861 and 1873. The classical groups, G2 and E6 É In his PhD thesis Leonard Eugene Dickson extended Jordan’s work to linear groups over all finite fields and included the unitary groups. -
Symmetric Generation and Existence of J3:2, the Automorphism Group of the Third Janko Group
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 304 (2006) 256–270 www.elsevier.com/locate/jalgebra Symmetric generation and existence of J3:2, the automorphism group of the third Janko group J.D. Bradley, R.T. Curtis ∗ School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 27 June 2005 Available online 28 February 2006 Communicated by Jan Saxl Abstract We make use of the primitive permutation action of degree 120 of the automorphism group of L2(16), the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J3 and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J3:2, to obtain the order of J3 and to prove its simplicity. A presentation for J3:2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J3:2. © 2006 Elsevier Inc. All rights reserved. 1. Introduction The group J3 was predicted by Janko in a paper in 1968 [10]. He considered a finite simple group with a single class of involutions in which the centraliser of an involution 1+4 was isomorphic to 2 :A5. The following year Higman and McKay, using an observation of Thompson that J3 must contain a subgroup isomorphic to L2(16):2, constructed the group on a computer [9]. -
RIGID LOCAL SYSTEMS with MONODROMY GROUP the CONWAY GROUP Co2
RIGID LOCAL SYSTEMS WITH MONODROMY GROUP THE CONWAY GROUP Co2 NICHOLAS M. KATZ, ANTONIO ROJAS-LEON,´ AND PHAM HUU TIEP Abstract. We first develop some basic facts about hypergeometric sheaves on the multi- plicative group Gm in characteristic p > 0. Certain of their Kummer pullbacks extend to irreducible local systems on the affine line in characteristic p > 0. One of these, of rank 23 in characteristic p = 3, turns out to have the Conway group Co2, in its irreducible orthogonal representation of degree 23, as its arithmetic and geometric monodromy groups. Contents Introduction 1 1. The basic set up, and general results 1 2. The criterion for finite monodromy 10 3. Theorems of finite monodromy 11 4. Determination of the monodromy groups 17 References 18 Introduction In the first two sections, we give the general set up. In the third section, we apply known criteria to show that certain local systems of rank 23 over the affine line in characteristic 3 have finite (arithmetic and geometric) monodromy groups. In the final section, we show that the finite monodromy groups in question are the Conway group Co2 in its 23-dimensional irreducible orthogonal representation. 1. The basic set up, and general results × We fix a prime number p, a prime number ` 6= p, and a nontrivial Q` -valued additive character of Fp. For k=Fp a finite extension, we denote by k the nontrivial additive character of k given by k := ◦ Tracek=Fp . In perhaps more down to earth terms, we fix a × nontrivial Q(µp) -valued additive character of Fp, and a field embedding of Q(µp) into Q` for some ` 6= p. -
The Cohomologies of the Sylow 2-Subgroups of a Symplectic Group of Degree Six and of the Third Conway Group
THE COHOMOLOGIES OF THE SYLOW 2-SUBGROUPS OF A SYMPLECTIC GROUP OF DEGREE SIX AND OF THE THIRD CONWAY GROUP JOHN MAGINNIS Abstract. The third Conway group Co3 is one of the twenty-six sporadic finite simple groups. The cohomology of its Sylow 2-subgroup S is computed, an important step in calculating the mod 2 cohomology of Co3. The spectral sequence for the central extension of S is described and collapses at the sixth page. Generators are described in terms of the Evens norm or transfers from subgroups. The central quotient S0 = S=2 is the Sylow 2-subgroup of the symplectic group Sp6(F2) of six by six matrices over the field of two elements. The cohomology of S0 is computed, and is detected by restriction to elementary abelian 2-subgroups. 1. Introduction A presentation for the Sylow 2-subgroup S of the sporadic finite simple group Co3 is given by Benson [1]. We will use a different presentation, with generators which can be described in terms of Benson's generators fa1; a2; a3; a4; b1; b2; c1; c2; eg as follows. w = c3; x = b2c1c2; y = ec3; z = a4; a = a3a4c2; b = a3a4b1c2; c = a3; t = a4c1; u = a2; v = a1: We have the following set of relations for the generators fw; x; y; z; a; b; c; t; u; vg. w2 = x2 = y2 = z2 = c2 = u2 = v2 = [w; y] = [w; z] = [w; c] = [w; v] = [x; z] = [x; u] = [x; v] = [y; c] = [y; u] = [y; v] = [a; z] = [c; z] = [u; z] = [v; z] = [c; t] = [c; u] = [c; v] = [a; u] = [a; v] = [b; c] = [b; t] = [b; u] = [b; v] = [t; u] = [t; v] = [u; v] = 1; tu = [a; y]; t = [w; b]; av = [w; x]; b = [x; y]; c = [y; z] u = b2 = [x; c] = [b; z] = [b; y] = [b; x]; uv = [t; x] = [a; b]; v = t2 = a2 = [w; u] = [a; c] = [t; z] = [a; w] = [t; w] = [t; a] = [t; y] = [a; x]: The group S has order 1024 and a center Z(S) of order two, generated by v = a1. -
Near Polygons and Fischer Spaces
Near polygons and Fischer spaces Citation for published version (APA): Brouwer, A. E., Cohen, A. M., Hall, J. I., & Wilbrink, H. A. (1994). Near polygons and Fischer spaces. Geometriae Dedicata, 49(3), 349-368. https://doi.org/10.1007/BF01264034 DOI: 10.1007/BF01264034 Document status and date: Published: 01/01/1994 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
The Moonshine Module for Conway's Group
Forum of Mathematics, Sigma (2015), Vol. 3, e10, 52 pages 1 doi:10.1017/fms.2015.7 THE MOONSHINE MODULE FOR CONWAY’S GROUP JOHN F. R. DUNCAN and SANDER MACK-CRANE Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA; email: [email protected], [email protected] Received 26 September 2014; accepted 30 April 2015 Abstract We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise. 2010 Mathematics Subject Classification: 11F11, 11F22, 17B69, 20C34 1. Introduction Taking the upper half-plane H τ C (τ/ > 0 , together with the Poincare´ 2 2 2 2 VD f 2 j = g metric ds y− .dx dy /, we obtain the Poincare´ half-plane model of the D C hyperbolic plane. -
Recent Progress in the Symmetric Generation of Groups, to Appear in the Proceedings of the Conference Groups St Andrews 2009 [28] B
RECENT PROGRESS IN THE SYMMETRIC GENERATION OF GROUPS BEN FAIRBAIRN Abstract. Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial struc- ture. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and prac- tically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmet- ric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic simple groups. ADDENDUM: This is an updated version of a survey article orig- inally accepted for inclusion in the proceedings of the 2009 ‘Groups St Andrews’ conference [27]. Since [27] was accepted the author has become aware of other recent work in the subject that we incorporate to provide an updated version here (the most notable addition being the contents of Section 3.4). 1. Introduction This article is concerned with groups that are generated by highly symmetric subsets of their elements: that is to say by subsets of el- ements whose set normalizer within the group they generate acts on them by conjugation in a highly symmetric manner. Rather than inves- tigate the behaviour of known groups we turn this procedure around and ask what groups can be generated by a set of elements that pos- sesses a certain assigned set of symmetries. This enables constructions arXiv:1004.2272v2 [math.GR] 21 Apr 2010 by hand of a number of interesting groups, including many of the spo- radic simple groups. Much of the emphasis of the research project to date has been concerned with using these techniques to construct spo- radic simple groups, and this article will emphasize this important spe- cial case. -
Quadratic Forms and Their Applications
Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999.