The Spirit of Moonshine: Connections Between the Mathieu Groups and Modular Forms
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Generating the Mathieu Groups and Associated Steiner Systems
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 112 (1993) 41-47 41 North-Holland Generating the Mathieu groups and associated Steiner systems Marston Conder Department of Mathematics and Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand Received 13 July 1989 Revised 3 May 1991 Abstract Conder, M., Generating the Mathieu groups and associated Steiner systems, Discrete Mathematics 112 (1993) 41-47. With the aid of two coset diagrams which are easy to remember, it is shown how pairs of generators may be obtained for each of the Mathieu groups M,,, MIz, Mz2, M,, and Mz4, and also how it is then possible to use these generators to construct blocks of the associated Steiner systems S(4,5,1 l), S(5,6,12), S(3,6,22), S(4,7,23) and S(5,8,24) respectively. 1. Introduction Suppose you land yourself in 24-dimensional space, and are faced with the problem of finding the best way to pack spheres in a box. As is well known, what you really need for this is the Leech Lattice, but alas: you forgot to bring along your Miracle Octad Generator. You need to construct the Leech Lattice from scratch. This may not be so easy! But all is not lost: if you can somehow remember how to define the Mathieu group Mz4, you might be able to produce the blocks of a Steiner system S(5,8,24), and the rest can follow. In this paper it is shown how two coset diagrams (which are easy to remember) can be used to obtain pairs of generators for all five of the Mathieu groups M,,, M12, M22> Mz3 and Mz4, and also how blocks may be constructed from these for each of the Steiner systems S(4,5,1 I), S(5,6,12), S(3,6,22), S(4,7,23) and S(5,8,24) respectively. -
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 ............................ -
12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups
12.6 Further Topics on Simple groups 387 12.6 Further Topics on Simple Groups This Web Section has three parts (a), (b) and (c). Part (a) gives a brief descriptions of the 56 (isomorphism classes of) simple groups of order less than 106, part (b) provides a second proof of the simplicity of the linear groups Ln(q), and part (c) discusses an ingenious method for constructing a version of the Steiner system S(5, 6, 12) from which several versions of S(4, 5, 11), the system for M11, can be computed. 12.6(a) Simple Groups of Order less than 106 The table below and the notes on the following five pages lists the basic facts concerning the non-Abelian simple groups of order less than 106. Further details are given in the Atlas (1985), note that some of the most interesting and important groups, for example the Mathieu group M24, have orders in excess of 108 and in many cases considerably more. Simple Order Prime Schur Outer Min Simple Order Prime Schur Outer Min group factor multi. auto. simple or group factor multi. auto. simple or count group group N-group count group group N-group ? A5 60 4 C2 C2 m-s L2(73) 194472 7 C2 C2 m-s ? 2 A6 360 6 C6 C2 N-g L2(79) 246480 8 C2 C2 N-g A7 2520 7 C6 C2 N-g L2(64) 262080 11 hei C6 N-g ? A8 20160 10 C2 C2 - L2(81) 265680 10 C2 C2 × C4 N-g A9 181440 12 C2 C2 - L2(83) 285852 6 C2 C2 m-s ? L2(4) 60 4 C2 C2 m-s L2(89) 352440 8 C2 C2 N-g ? L2(5) 60 4 C2 C2 m-s L2(97) 456288 9 C2 C2 m-s ? L2(7) 168 5 C2 C2 m-s L2(101) 515100 7 C2 C2 N-g ? 2 L2(9) 360 6 C6 C2 N-g L2(103) 546312 7 C2 C2 m-s L2(8) 504 6 C2 C3 m-s -
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. -
On Quadruply Transitive Groups Dissertation
ON QUADRUPLY TRANSITIVE GROUPS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By ERNEST TILDEN PARKER, B. A. The Ohio State University 1957 Approved by: 'n^CLh^LaJUl 4) < Adviser Department of Mathematics Acknowledgment The author wishes to express his sincere gratitude to Professor Marshall Hall, Jr., for assistance and encouragement in the preparation of this dissertation. ii Table of Contents Page 1. Introduction ------------------------------ 1 2. Preliminary Theorems -------------------- 3 3. The Main Theorem-------------------------- 12 h. Special Cases -------------------------- 17 5. References ------------------------------ kZ iii Introduction In Section 3 the following theorem is proved: If G is a quadruplv transitive finite permutation group, H is the largest subgroup of G fixing four letters, P is a Sylow p-subgroup of H, P fixes r & 1 2 letters and the normalizer in G of P has its transitive constituent Aj. or Sr on the letters fixed by P, and P has no transitive constituent of degree ^ p3 and no set of r(r-l)/2 similar transitive constituents, then G is. alternating or symmetric. The corollary following the theorem is the main result of this dissertation. 'While less general than the theorem, it provides arithmetic restrictions on primes dividing the order of the sub group fixing four letters of a quadruply transitive group, and on the degrees of Sylow subgroups. The corollary is: ■ SL G is. a quadruplv transitive permutation group of degree n - kp+r, with p prime, k<p^, k<r(r-l)/2, rfc!2, and the subgroup of G fixing four letters has a Sylow p-subgroup P of degree kp, and the normalizer in G of P has its transitive constituent A,, or Sr on the letters fixed by P, then G is. -
Arxiv:1911.10534V3 [Math.AT] 17 Apr 2020 Statement
THE ANDO-HOPKINS-REZK ORIENTATION IS SURJECTIVE SANATH DEVALAPURKAR Abstract. We show that the map π∗MString ! π∗tmf induced by the Ando-Hopkins-Rezk orientation is surjective. This proves an unpublished claim of Hopkins and Mahowald. We do so by constructing an E1-ring B and a map B ! MString such that the composite B ! MString ! tmf is surjective on homotopy. Applications to differential topology, and in particular to Hirzebruch's prize question, are discussed. 1. Introduction The goal of this paper is to show the following result. Theorem 1.1. The map π∗MString ! π∗tmf induced by the Ando-Hopkins-Rezk orientation is surjective. This integral result was originally stated as [Hop02, Theorem 6.25], but, to the best of our knowledge, no proof has appeared in the literature. In [HM02], Hopkins and Mahowald give a proof sketch of Theorem 1.1 for elements of π∗tmf of Adams-Novikov filtration 0. The analogue of Theorem 1.1 for bo (namely, the statement that the map π∗MSpin ! π∗bo induced by the Atiyah-Bott-Shapiro orientation is surjective) is classical [Mil63]. In Section2, we present (as a warmup) a proof of this surjectivity result for bo via a technique which generalizes to prove Theorem 1.1. We construct an E1-ring A with an E1-map A ! MSpin. The E1-ring A is a particular E1-Thom spectrum whose mod 2 homology is given by the polynomial subalgebra 4 F2[ζ1 ] of the mod 2 dual Steenrod algebra. The Atiyah-Bott-Shapiro orientation MSpin ! bo is an E1-map, and so the composite A ! MSpin ! bo is an E1-map. -
Our Mathematical Universe: I. How the Monster Group Dictates All of Physics
October, 2011 PROGRESS IN PHYSICS Volume 4 Our Mathematical Universe: I. How the Monster Group Dictates All of Physics Franklin Potter Sciencegems.com, 8642 Marvale Drive, Huntington Beach, CA 92646. E-mail: [email protected] A 4th family b’ quark would confirm that our physical Universe is mathematical and is discrete at the Planck scale. I explain how the Fischer-Greiss Monster Group dic- tates the Standard Model of leptons and quarks in discrete 4-D internal symmetry space and, combined with discrete 4-D spacetime, uniquely produces the finite group Weyl E8 x Weyl E8 = “Weyl” SO(9,1). The Monster’s j-invariant function determines mass ratios of the particles in finite binary rotational subgroups of the Standard Model gauge group, dictates Mobius¨ transformations that lead to the conservation laws, and connects interactions to triality, the Leech lattice, and Golay-24 information coding. 1 Introduction 5. Both 4-D spacetime and 4-D internal symmetry space are discrete at the Planck scale, and both spaces can The ultimate idea that our physical Universe is mathematical be telescoped upwards mathematically by icosians to at the fundamental scale has been conjectured for many cen- 8-D spaces that uniquely combine into 10-D discrete turies. In the past, our marginal understanding of the origin spacetime with discrete Weyl E x Weyl E symmetry of the physical rules of the Universe has been peppered with 8 8 (not the E x E Lie group of superstrings/M-theory). huge gaps, but today our increased understanding of funda- 8 8 mental particles promises to eliminate most of those gaps to 6. -
Symmetries and Exotic Smooth Structures on a K3 Surface WM Chen University of Massachusetts - Amherst, [email protected]
University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Mathematics and Statistics Publication Series 2008 Symmetries and exotic smooth structures on a K3 surface WM Chen University of Massachusetts - Amherst, [email protected] S Kwasik Follow this and additional works at: https://scholarworks.umass.edu/math_faculty_pubs Part of the Physical Sciences and Mathematics Commons Recommended Citation Chen, WM and Kwasik, S, "Symmetries and exotic smooth structures on a K3 surface" (2008). JOURNAL OF TOPOLOGY. 268. Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/268 This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. SYMMETRIES AND EXOTIC SMOOTH STRUCTURES ON A K3 SURFACE WEIMIN CHEN AND SLAWOMIR KWASIK Abstract. Smooth and symplectic symmetries of an infinite family of distinct ex- otic K3 surfaces are studied, and comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order ≥ 7 is proved and nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order ≥ 5 is explicitly determined, provided that the action is homologically nontrivial. -
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. -
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.......................... -
§2. Elliptic Curves: J-Invariant (Jan 31, Feb 4,7,9,11,14) After
24 JENIA TEVELEV §2. Elliptic curves: j-invariant (Jan 31, Feb 4,7,9,11,14) After the projective line P1, the easiest algebraic curve to understand is an elliptic curve (Riemann surface of genus 1). Let M = isom. classes of elliptic curves . 1 { } We are going to assign to each elliptic curve a number, called its j-invariant and prove that 1 M1 = Aj . 1 1 So as a space M1 A is not very interesting. However, understanding A ! as a moduli space of elliptic curves leads to some breath-taking mathemat- ics. More generally, we introduce M = isom. classes of smooth projective curves of genus g g { } and M = isom. classes of curves C of genus g with points p , . , p C . g,n { 1 n ∈ } We will return to these moduli spaces later in the course. But first let us recall some basic facts about algebraic curves = compact Riemann surfaces. We refer to [G] and [Mi] for a rigorous and detailed exposition. §2.1. Algebraic functions, algebraic curves, and Riemann surfaces. The theory of algebraic curves has roots in analysis of Abelian integrals. An easiest example is the elliptic integral: in 1655 Wallis began to study the arc length of an ellipse (X/a)2 + (Y/b)2 = 1. The equation for the ellipse can be solved for Y : Y = (b/a) (a2 X2), − and this can easily be differentiated !to find bX Y ! = − . a√a2 X2 − 2 This is squared and put into the integral 1 + (Y !) dX for the arc length. Now the substitution x = X/a results in " ! 1 e2x2 s = a − dx, 1 x2 # $ − between the limits 0 and X/a, where e = 1 (b/a)2 is the eccentricity. -
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.