The Radical Subgroups of the Fischer Simple Groups
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Atlasrep —A GAP 4 Package
AtlasRep —A GAP 4 Package (Version 2.1.0) Robert A. Wilson Richard A. Parker Simon Nickerson John N. Bray Thomas Breuer Robert A. Wilson Email: [email protected] Homepage: http://www.maths.qmw.ac.uk/~raw Richard A. Parker Email: [email protected] Simon Nickerson Homepage: http://nickerson.org.uk/groups John N. Bray Email: [email protected] Homepage: http://www.maths.qmw.ac.uk/~jnb Thomas Breuer Email: [email protected] Homepage: http://www.math.rwth-aachen.de/~Thomas.Breuer AtlasRep — A GAP 4 Package 2 Copyright © 2002–2019 This package may be distributed under the terms and conditions of the GNU Public License Version 3 or later, see http://www.gnu.org/licenses. Contents 1 Introduction to the AtlasRep Package5 1.1 The ATLAS of Group Representations.........................5 1.2 The GAP Interface to the ATLAS of Group Representations..............6 1.3 What’s New in AtlasRep, Compared to Older Versions?...............6 1.4 Acknowledgements................................... 14 2 Tutorial for the AtlasRep Package 15 2.1 Accessing a Specific Group in AtlasRep ........................ 16 2.2 Accessing Specific Generators in AtlasRep ...................... 18 2.3 Basic Concepts used in AtlasRep ........................... 19 2.4 Examples of Using the AtlasRep Package....................... 21 3 The User Interface of the AtlasRep Package 33 3.1 Accessing vs. Constructing Representations...................... 33 3.2 Group Names Used in the AtlasRep Package..................... 33 3.3 Standard Generators Used in the AtlasRep Package.................. 34 3.4 Class Names Used in the AtlasRep Package...................... 34 3.5 Accessing Data via AtlasRep ............................ -
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. -
Classifying Spaces, Virasoro Equivariant Bundles, Elliptic Cohomology and Moonshine
CLASSIFYING SPACES, VIRASORO EQUIVARIANT BUNDLES, ELLIPTIC COHOMOLOGY AND MOONSHINE ANDREW BAKER & CHARLES THOMAS Introduction This work explores some connections between the elliptic cohomology of classifying spaces for finite groups, Virasoro equivariant bundles over their loop spaces and Moonshine for finite groups. Our motivation is as follows: up to homotopy we can replace the loop group LBG by ` the disjoint union [γ] BCG(γ) of classifying spaces of centralizers of elements γ representing conjugacy classes of elements in G. An elliptic object over LBG then becomes a compatible family of graded infinite dimensional representations of the subgroups CG(γ), which in turn E ∗ defines an element in J. Devoto's equivariant elliptic cohomology ring ``G. Up to localization (inversion of the order of G) and completion with respect to powers of the kernel of the E ∗ −! E ∗ ∗ homomorphism ``G ``f1g, this ring is isomorphic to E`` (BG), see [5, 6]. Moreover, elliptic objects of this kind are already known: for example the 2-variable Thompson series forming part of the Moonshine associated with the simple groups M24 and the Monster M. Indeed the compatibility condition between the characters of the representations of CG(γ) mentioned above was originally formulated by S. Norton in an Appendix to [21], independently of the work on elliptic genera by various authors leading to the definition of elliptic cohom- ology (see the various contributions to [17]). In a slightly different direction, J-L. Brylinski [2] introduced the group of Virasoro equivariant bundles over the loop space LM of a simply connected manifold M as part of his investigation of a Dirac operator on LM with coefficients in a suitable vector bundle. -
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.......................... -
K3 Surfaces, N= 4 Dyons, and the Mathieu Group
K3 Surfaces, =4 Dyons, N and the Mathieu Group M24 Miranda C. N. Cheng Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural inter- pretation in terms of the dimensions of representations of the largest Mathieu group M24. In this paper we first give further evidence for this possibility by studying the elliptic genus of K3 surfaces twisted by some simple symplectic automorphisms. These partition functions with insertions of elements of M24 (the McKay-Thompson series) give further information about the relevant representation. We then point out that this new “moonshine” for the largest Mathieu group is con- nected to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of K3 T 2-compactified type II string theory. This insight on the symmetry× of the theory sheds new light on the gener- alised Kac-Moody algebra structure appearing in the spectrum, and leads to predictions for new elliptic genera of K3, perturbative spec- arXiv:1005.5415v2 [hep-th] 3 Jun 2010 trum of the toroidally compactified heterotic string, and the index for the 1/4-BPS dyons in the d = 4, = 4 string theory, twisted by elements of the group of stringy K3N isometries. 1 1 Introduction and Summary Recently there have been two new observations relating K3 surfaces and the largest Mathieu group M24. They seem to suggest that the sporadic group M24 naturally acts on the spectrum of K3-compactified string theory. -
THE FISCHER-CLIFFORD MATRICES of a MAXIMAL SUBGROUP of Fi24 1. Introduction Let ¯G = N·G Be an Extension of N by G, Where N Is
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 7, Pages 300{321 (July 29, 2003) S 1088-4165(03)00175-4 THE FISCHER-CLIFFORD MATRICES 0 OF A MAXIMAL SUBGROUP OF Fi24 FARYAD ALI AND JAMSHID MOORI Abstract. 0 The Fischer group Fi24 is the largest sporadic simple Fischer group of order 1255205709190661721292800 = 221:316:52:73:11:13:17:23:29 : 0 The group Fi24 is the derived subgroup of the Fischer 3-transposition group Fi24 discovered by Bernd Fischer. There are five classes of elements of order 0 3inFi24 as represented in ATLAS by 3A,3B,3C,3D and 3E. A subgroup 0 2f g of Fi24 of order 3 is called of type 3X,whereX A; B; C; D; E ,ifitisgen- erated by an element in the class 3X. There are six classes of maximal 3-local 0 subgroups of Fi24 as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local ¯ h i ∼ 7· ∼ 7 subgroups G := NFi0 ( N ) = 3 O7(3), where N = 3 is the natural orthog- 24∼ onal module for G=N¯ = O7(3) with 364 subgroups of type 3B corresponding to the totally isotropic points. The group G¯ is a nonsplit extension of N by ∼ G = O7(3). 1. Introduction Let G¯ = N·G be an extension of N by G,whereN is a normal subgroup of G¯ ∼ and G=N¯ = G.Let¯g 2 G¯ be a lifting of g 2 G under the natural homomorphism G¯ −→ G and let [g] be a conjugacy class of elements of G with representative 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. -
THERE IS NO Sz(8) in the MONSTER 1. Introduction The
THERE IS NO Sz(8) IN THE MONSTER ROBERT A. WILSON Abstract. As a contribution to an eventual solution of the problem of the determination of the maximal subgroups of the Monster we show that there is no subgroup isomorphic to Sz(8). This also completes the determination of exactly which simple groups are subgroups of which of the 26 sporadic simple groups. The proof is largely, though not entirely, computer-free. 1. Introduction The Fischer{Griess Monster group M is the largest of the 26 sporadic simple groups, and was first constructed by Griess [6] in 1982. A simplified construction along the same general lines was given by Conway [2]. One of the major problems in group theory today is that of classifying the max- imal subgroups of the finite simple groups and their automorphism groups (see, for example, [1]). Much work has been done over many years attempting to determine the maximal subgroups of M, but it is still the only sporadic group whose maximal subgroups are not completely classified (see [23] and references therein). The maximal p-local subgroups of the Monster were classified in [22, 15, 16], and much theoretical work on non-local subgroups was accomplished in [17, 18]. Following successful computer constructions of the Monster [14, 8] other techniques became available, and further progress was made [9, 11, 7, 19, 26, 27], including discovery of five previously unknown maximal subgroups, isomorphic to PSL2(71), PSL2(59), PSL2(41), PGL2(29), PGL2(19). The cases left open by this previous work are possible maximal subgroups with socle isomorphic to one of the following simple groups: PSL2(8); PSL2(13); PSL2(16); PSU3(4); PSU3(8); Sz(8): Of these, PSL2(8) and PSL2(16) have been classified in unpublished work of P. -
The Mathieu Group M24 As We Knew It
Cambridge University Press 978-1-108-42978-8 — The Mathieu Groups A. A. Ivanov Excerpt More Information 1 The Mathieu Group M24 As We Knew It In this chapter we start by reviewing the common way to describe the Math- ieu group G = M24 as the automorphism group of the binary Golay code and of the Steiner system formed by the minimal codewords in the Golay code. This discussion will lead us to the structure of the octad–trio–sextet stabiliz- ers {G1, G2, G3}. The starting point is the observation that, when forming a similar triple {H1, H2, H3} comprised of the stabilizers of one-, two- and three-dimensional subspaces in H = L5(2),wehave 4 6 G1 =∼ H1 =∼ 2 : L4(2), G2 =∼ H2 =∼ 2 : (L3(2) × S3), [G1 : G1 ∩ G2]=[H1 : H1 ∩ H2]=15, [G2 : G1 ∩ G2]=[H2 : H1 ∩ H1]=3, so that the amalgams {G1, G2} and {H1, H2} have the same type according to Goldschmidt’s terminology, although they are not isomorphic and differ by a twist performed by an outer automorphism of 3 3 H1 ∩ H2 =∼ G1 ∩ G2 =∼ (2 × 2 ) : (L3(2) × 2), which permutes conjugacy classes of L3(2)-subgroups. This observation led us to the construction of M24 as the universal completion of a twisted L5(2)- amalgam. 1.1 The Golay Code Commonly the Mathieu group M24 is defined as the automorphism group of the Golay code, which by definition is a (a) binary (b) linear code (c) of length 24, which is (d) even, (e) self-dual and (f) has no codewords of weight 4. -
The Symmetric Genus of the Fischer Group Fi23
The symmetric genus of the Fischer group F i23 Robert A. Wilson School of Mathematics and Statistics, The University of Birmingham, England published in Topology 36 (1996), 379{380 Abstract We show that the sporadic simple group F i23 is generated by an element of order 2 and an element of order 3, whose product has order 8. Since F i23 is not a Hurwitz group, we can deduce the symmetric genus of the group. In [4] the symmetric genus of a finite group G is defined to be the smallest integer g such that G acts faithfully on a closed orientable surface of genus g. The Riemann{Hurwitz formula implies that this minimum is attained by minimising n 1 X 1 − i=1 ai n Q ai where G is generated by elements xi satisfying xj = 1 and xi = 1 for all j=1 i. In most, but not all, cases, this minimum is attained for n = 3, and we have 1 1 1 1 g = 1 + jGj(1 − − − ); 2 r s t where 1 1 1 + + r s t is maximal subject to the existence of elements x, y and z generating G, with xr = ys = zt = xyz = 1: 1 The symmetric genus of each of the sporadic simple groups is discussed in [2], where a table of results shows that the symmetric genus is now known in 23 of the 26 cases. The remaining three are F i23, B and M. The case of the Baby Monster was dealt with in [5]. In this paper we deal with the case F i23. -
Finite Groups Acting on Homology Manifolds
pacific journal of mathematics Vol. 181, No. 3, 1997 FINITE GROUPS ACTING ON HOMOLOGY MANIFOLDS Michael Aschbacher To the memory of Olga Taussky-Todd 1. Introduction. In this paper we study homology manifolds T admitting the action of a fi- nite group preserving the structure of a regular CW-complex on T . The CW-complex is parameterized by a poset and the topological properties of the manifold are translated into a combinatorial setting via the poset. We concentrate on n-manifolds which admit a fairly rigid group of automor- phisms transitive on the n-cells of the complex. This allows us to make yet another translation from a combinatorial into a group theoretic setting. We close by using our machinery to construct representations on manifolds of the Monster, the largest sporadic group. Some of these manifolds are of dimension 24, and hence candidates for examples to Hirzebruch’s Prize Question in [HBJ], but unfortunately closer inspection shows the Aˆ-genus of these manifolds is 0 rather than 1, so none is a Hirzebruch manifold. In order to state our results precisely we need to recall some definitions and introduce a few concepts. The geometric realization functor T associates to each poset X a topological space T (X) and to each x ∈ X a closed subspace T (x)ofT(X). The cells T (x), x ∈ X, supply a cell structure on T (X) parametrized by X. In addition there is a simplicial complex O(X) associated to X called the order complex of X and a canonical triangulation ϕ : O(X) → T (X)ofT(X)byO(X). -
On Rademacher Sums, the Largest Mathieu Group and the Holographic Modularity of Moonshine Miranda C.N
communications in number theory and physics Volume 6, Number 3, 697–758, 2012 On Rademacher sums, the largest Mathieu group and the holographic modularity of moonshine Miranda C.N. Cheng and John F.R. Duncan Recently a conjecture has proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero prop- erty: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invari- ance groups define genus zero quotients of the upper half-plane. It is therefore natural to ask if there is an analogue of this property in the Mathieu case, and at first glance the answer appears to be negative since not all the discrete groups arising there have genus zero. On the other hand, in this paper we prove that each (mock) modular form appearing in the Mathieu correspondence coincides with the Rademacher sum constructed from its polar part. This property, inspired by the AdS/CFT correspondence in physics, was shown previously to be equivalent to the genus zero property of monstrous moonshine. Hence, we conclude that this “Rademacher summability” property serves as the natural analogue of the genus zero property in the Mathieu case. Our result constitutes further evidence that the Rademacher method provides a powerful frame- work for understanding the modularity of moonshine, and leads to interesting physical questions regarding the gravitational duals of the relevant conformal field theories.