My Life and Times with the Simple Sporadic Groups
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Math 411 Midterm 2, Thursday 11/17/11, 7PM-8:30PM. Instructions: Exam Time Is 90 Mins
Math 411 Midterm 2, Thursday 11/17/11, 7PM-8:30PM. Instructions: Exam time is 90 mins. There are 7 questions for a total of 75 points. Calculators, notes, and textbook are not allowed. Justify all your answers carefully. If you use a result proved in the textbook or class notes, state the result precisely. 1 Q1. (10 points) Let σ 2 S9 be the permutation 1 2 3 4 5 6 7 8 9 5 8 7 2 3 9 1 4 6 (a) (2 points) Write σ as a product of disjoint cycles. (b) (2 points) What is the order of σ? (c) (2 points) Write σ as a product of transpositions. (d) (4 points) Compute σ100. Q2. (8 points) (a) (3 points) Find an element of S5 of order 6 or prove that no such element exists. (b) (5 points) Find an element of S6 of order 7 or prove that no such element exists. Q3. (12 points) (a) (4 points) List all the elements of the alternating group A4. (b) (8 points) Find the left cosets of the subgroup H of A4 given by H = fe; (12)(34); (13)(24); (14)(23)g: Q4. (10 points) (a) (3 points) State Lagrange's theorem. (b) (7 points) Let p be a prime, p ≥ 3. Let Dp be the dihedral group of symmetries of a regular p-gon (a polygon with p sides of equal length). What are the possible orders of subgroups of Dp? Give an example in each case. 2 Q5. (13 points) (a) (3 points) State a theorem which describes all finitely generated abelian groups. -
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. -
Classification of Finite Abelian Groups
Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k. -
Mathematics 310 Examination 1 Answers 1. (10 Points) Let G Be A
Mathematics 310 Examination 1 Answers 1. (10 points) Let G be a group, and let x be an element of G. Finish the following definition: The order of x is ... Answer: . the smallest positive integer n so that xn = e. 2. (10 points) State Lagrange’s Theorem. Answer: If G is a finite group, and H is a subgroup of G, then o(H)|o(G). 3. (10 points) Let ( a 0! ) H = : a, b ∈ Z, ab 6= 0 . 0 b Is H a group with the binary operation of matrix multiplication? Be sure to explain your answer fully. 2 0! 1/2 0 ! Answer: This is not a group. The inverse of the matrix is , which is not 0 2 0 1/2 in H. 4. (20 points) Suppose that G1 and G2 are groups, and φ : G1 → G2 is a homomorphism. (a) Recall that we defined φ(G1) = {φ(g1): g1 ∈ G1}. Show that φ(G1) is a subgroup of G2. −1 (b) Suppose that H2 is a subgroup of G2. Recall that we defined φ (H2) = {g1 ∈ G1 : −1 φ(g1) ∈ H2}. Prove that φ (H2) is a subgroup of G1. Answer:(a) Pick x, y ∈ φ(G1). Then we can write x = φ(a) and y = φ(b), with a, b ∈ G1. Because G1 is closed under the group operation, we know that ab ∈ G1. Because φ is a homomorphism, we know that xy = φ(a)φ(b) = φ(ab), and therefore xy ∈ φ(G1). That shows that φ(G1) is closed under the group operation. -
A Class of Frobenius Groups
A CLASS OF FROBENIUS GROUPS DANIEL GORKNSTEIN 1. Introduction. If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a' in A, b 5* 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order. A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transforma tion x' = cox, where co is a primitive element of K, and by b the transformation x' = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively. Since G admits a faithful representation on m letters {m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group. In Theorem I we shall show that this property is characteristic of independent ABA-groups. In a Frobenius group on m letters, the set of elements whose order divides m forms a normal subgroup, called the regular subgroup. In our example, the regular subgroup M of G consists of the set of translations, and hence is an Abelian group of order m = pn and of type (p, p, . -
LMS – EPSRC Durham Symposium
LMS – EPSRC Durham Symposium Anthony Byrne Grants and Membership Administrator 12th July 2016, Durham The work of the LMS for mathematics The charitable aims of the Society: Funding the advancement of mathematical knowledge Encouraging mathematical research and collaboration ’, George Legendre Celebrating mathematical 30 Pieces achievements Publishing and disseminating mathematical knowledge Advancing and promoting mathematics The attendees of the Young Researchers in Mathematics Conference 2015, held at Oxford Historical Moments of the London Mathematical Society 1865 Foundation of LMS at University College London George Campbell De Morgan organised the first meeting, and his father, Augustus De Morgan became the 1st President 1865 First minute book list of the 27 original members 1866 LMS moves to Old Burlington House, Piccadilly J.J. Sylvester, 2nd President of the Society. 1866 Julius Plûcker Thomas Hirst Plûcker Collection of boxwood models of quartic surfaces given to Thomas Archer Hirst, Vice- President of LMS, and donated to the Society 1870 Move to Asiatic Society, 22 Albemarle Street William Spottiswoode, President 1874 Donation of £1,000 from John William Strutt (Lord Rayleigh) Generous donation enabled the Society to publish volumes of the Proceedings of the London Mathematical Society. J.W. Strutt (Lord Rayleigh), LMS President 1876-78 1881 First women members Charlotte Angas Scott and Christine Ladd 1884 First De Morgan medal awarded to Arthur Cayley 1885 Sophie Bryant First woman to have a paper published in LMS Proceedings 1916 Return to Burlington House the home of LMS until 1998 1937 ACE ’s Automatic Turing LMS Proceedings, 1937 Computing Engine, published Alan Turing’s first paper 1950 On Computable Numbers 1947 Death of G.H. -
Density of Algebraic Points on Noetherian Varieties 3
DENSITY OF ALGEBRAIC POINTS ON NOETHERIAN VARIETIES GAL BINYAMINI Abstract. Let Ω ⊂ Rn be a relatively compact domain. A finite collection of real-valued functions on Ω is called a Noetherian chain if the partial derivatives of each function are expressible as polynomials in the functions. A Noether- ian function is a polynomial combination of elements of a Noetherian chain. We introduce Noetherian parameters (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila-Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any ε> 0, the number of points of height H in the tran- scendental part of the set is at most C ·Hε where C can be explicitly estimated from the Noetherian parameters and ε. We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular func- tions and universal covers of compact Riemann surfaces, Jacobi theta func- tions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety Ag . We thus effectivize the (geometric side of) Pila-Zannier strategy for unlikely intersections in those instances that involve only compact domains. 1. Introduction 1.1. The (real) Noetherian class. Let ΩR ⊂ Rn be a bounded domain, and n denote by x := (x1,...,xn) a system of coordinates on R . A collection of analytic ℓ functions φ := (φ1,...,φℓ): Ω¯ R → R is called a (complex) real Noetherian chain if it satisfies an overdetermined system of algebraic partial differential equations, i =1,...,ℓ ∂φi = Pi,j (x, φ), (1) ∂xj j =1,...,n where P are polynomials. -
A Note on Frobenius Groups
Journal of Algebra 228, 367᎐376Ž. 2000 doi:10.1006rjabr.2000.8269, available online at http:rrwww.idealibrary.com on A Note on Frobenius Groups Paul Flavell The School of Mathematics and Statistics, The Uni¨ersity of Birmingham, CORE Birmingham B15 2TT, United Kingdom Metadata, citation and similar papers at core.ac.uk E-mail: [email protected] Provided by Elsevier - Publisher Connector Communicated by George Glauberman Received January 25, 1999 1. INTRODUCTION Two celebrated applications of the character theory of finite groups are Burnside's p␣q -Theorem and the theorem of Frobenius on the groups that bear his name. The elegant proofs of these theorems were obtained at the beginning of this century. It was then a challenge to find character-free proofs of these results. This was achieved for the p␣q -Theorem by Benderwx 1 , Goldschmidt wx 3 , and Matsuyama wx 8 in the early 1970s. Their proofs used ideas developed by Feit and Thompson in their proof of the Odd Order Theorem. There is no known character-free proof of Frobe- nius' Theorem. Recently, CorradiÂÂ and Horvathwx 2 have obtained some partial results in this direction. The purpose of this note is to prove some stronger results. We hope that this will stimulate more interest in this problem. Throughout the remainder of this note, we let G be a finite Frobenius group. That is, G contains a subgroup H such that 1 / H / G and H l H g s 1 for all g g G y H. A subgroup with these properties is called a Frobenius complement of G. -
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 ............................ -
Irreducible Character Restrictions to Maximal Subgroups of Low-Rank Classical Groups of Type B and C
IRREDUCIBLE CHARACTER RESTRICTIONS TO MAXIMAL SUBGROUPS OF LOW-RANK CLASSICAL GROUPS OF TYPE B AND C KEMPTON ALBEE, MIKE BARNES, AARON PARKER, ERIC ROON, AND A.A. SCHAEFFER FRY Abstract Representations are special functions on groups that give us a way to study abstract groups using matrices, which are often easier to understand. In particular, we are often interested in irreducible representations, which can be thought of as the building blocks of all representations. Much of the information about these representations can then be understood by instead looking at the trace of the matrices, which we call the character of the representation. This paper will address restricting characters to subgroups by shrinking the domain of the original representation to just the subgroup. In particular, we will discuss the problem of determining when such restricted characters remain irreducible for certain low-rank classical groups. 1. Introduction Given a finite group G, a (complex) representation of G is a homomorphism Ψ: G ! GLn(C). By summing the diagonal entries of the images Ψ(g) for g 2 G (that is, taking the trace of the matrices), we obtain the corresponding character, χ = Tr◦Ψ of G. The degree of the representation Ψ or character χ is n = χ(1). It is well-known that any character of G can be written as a sum of so- called irreducible characters of G. In this sense, irreducible characters are of particular importance in representation theory, and we write Irr(G) to denote the set of irreducible characters of G. Given a subgroup H of G, we may view Ψ as a representation of H as well, simply by restricting the domain. -
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. -
The Top Mathematics Award
Fields told me and which I later verified in Sweden, namely, that Nobel hated the mathematician Mittag- Leffler and that mathematics would not be one of the do- mains in which the Nobel prizes would The Top Mathematics be available." Award Whatever the reason, Nobel had lit- tle esteem for mathematics. He was Florin Diacuy a practical man who ignored basic re- search. He never understood its impor- tance and long term consequences. But Fields did, and he meant to do his best John Charles Fields to promote it. Fields was born in Hamilton, Ontario in 1863. At the age of 21, he graduated from the University of Toronto Fields Medal with a B.A. in mathematics. Three years later, he fin- ished his Ph.D. at Johns Hopkins University and was then There is no Nobel Prize for mathematics. Its top award, appointed professor at Allegheny College in Pennsylvania, the Fields Medal, bears the name of a Canadian. where he taught from 1889 to 1892. But soon his dream In 1896, the Swedish inventor Al- of pursuing research faded away. North America was not fred Nobel died rich and famous. His ready to fund novel ideas in science. Then, an opportunity will provided for the establishment of to leave for Europe arose. a prize fund. Starting in 1901 the For the next 10 years, Fields studied in Paris and Berlin annual interest was awarded yearly with some of the best mathematicians of his time. Af- for the most important contributions ter feeling accomplished, he returned home|his country to physics, chemistry, physiology or needed him.