2020 Ural Workshop on Group Theory and Combinatorics
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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, . -
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. -
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. -
Graphs Based on Hoop Algebras
mathematics Article Graphs Based on Hoop Algebras Mona Aaly Kologani 1, Rajab Ali Borzooei 2 and Hee Sik Kim 3,* 1 Hatef Higher Education Institute, Zahedan 8301, Iran; [email protected] 2 Department of Mathematics, Shahid Beheshti University, Tehran 7561, Iran; [email protected] 3 Research Institute for Natural Science, Department of Mathematics, Hanyang University, Seoul 04763, Korea * Correspondence: [email protected]; Tel.: +82-2-2220-0897 Received: 14 March 2019; Accepted: 17 April 2019 Published: 21 April 2019 Abstract: In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using zero divisor elements, we construct a productive graph and prove that it is connected and both complete and a tree under some conditions. Keywords: hoop algebra; zero divisor; implicative graph; productive graph MSC: 06F99; 03G25; 18B35; 22A26 1. Introduction Non-classical logic has become a formal and useful tool for computer science to deal with uncertain information and fuzzy information. The algebraic counterparts of some non-classical logics satisfy residuation and those logics can be considered in a frame of residuated lattices [1]. For example, Hájek’s BL (basic logixc [2]), Lukasiewicz’s MV (many-valued logic [3]) and MTL (monoidal t-norm-based logic [4]) are determined by the class of BL-algebras, MV-algebras and MTL-algebras, respectively. All of these algebras have lattices with residuation as a common support set. -
Michael Borinsky Graphs in Perturbation Theory Algebraic Structure and Asymptotics Springer Theses
Springer Theses Recognizing Outstanding Ph.D. Research Michael Borinsky Graphs in Perturbation Theory Algebraic Structure and Asymptotics Springer Theses Recognizing Outstanding Ph.D. Research Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists. Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the signifi- cance of its content. -
Sporadic Sics and Exceptional Lie Algebras
Sporadic SICs and Exceptional Lie Algebras Blake C. Staceyy yQBism Group, Physics Department, University of Massachusetts Boston, 100 Morrissey Boulevard, Boston MA 02125, USA November 13, 2019 Abstract Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy. 1 Preliminaries A set of equiangular lines is a set of unit vectors in a d-dimensional vector space such that the magnitude of the inner product of any pair is constant: 1; j = k; jhv ; v ij = (1) j k α; j 6= k: The maximum number of equiangular lines in a space of dimension d (the so-called Gerzon bound) is d(d+1)=2 for real vector spaces and d2 for complex. In the real case, the Gerzon bound is only known to be attained in dimensions 2, 3, 7 and 23, and we know it can't be attained in general. If you like peculiar alignments of mathematical topics, the appearance of 7 and 23 might make your ears prick up here. If you made the wild guess that the octonions and the Leech lattice are just around the corner. you'd be absolutely right. Meanwhile, the complex case is of interest for quantum information theory, because a set of d2 equiangular lines in Cd specifies a measurement that can be performed upon a quantum-mechanical system. These measurements are highly symmetric, in that the lines which specify them are equiangular, and they are \informationally complete" in a sense that quantum theory makes precise. -
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 ............................ -
Lecture 14. Frobenius Groups (II)
Quick Review Proof of Frobenius’ Theorem Heisenberg groups Lecture 14. Frobenius Groups (II) Daniel Bump May 28, 2020 Quick Review Proof of Frobenius’ Theorem Heisenberg groups Review: Frobenius Groups Definition A Frobenius Group is a group G with a faithful transitive action on a set X such that no element fixes more than one point. An action of G on a set X gives a homomorphism from G to the group of bijections X (the symmetric group SjXj). In this definition faithful means this homomorphism is injective. Let H be the stabilizer of a point x0 2 X. The group H is called the Frobenius complement. Today we will prove: Theorem (Frobenius (1901)) A Frobenius group G is a semidirect product. That is, there exists a normal subgroup K such that G = HK and H \ K = f1g. Quick Review Proof of Frobenius’ Theorem Heisenberg groups Review: The mystery of Frobenius’ Theorem Since Frobenius’ theorem doesn’t require group representation theory in its formulation, it is remarkable that no proof has ever been found that doesn’t use representation theory! Web links: Frobenius groups (Wikipedia) Fourier Analytic Proof of Frobenius’ Theorem (Terence Tao) Math Overflow page on Frobenius’ theorem Frobenius Groups (I) (Lecture 14) Quick Review Proof of Frobenius’ Theorem Heisenberg groups The precise statement Last week we introduced the notion of a Frobenius group. This is a group G that acts transitively on a set X in which no element except the identity fixes more than one point. Let H be the isotropy subgroup of an element, and let be K∗ [ f1g where K∗ is the set of elements with no fixed points. -
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 -
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. -
Arxiv:2002.11183V2 [Math.AG]
Arithmetic statistics on cubic surfaces Ronno Das April 6, 2020 Abstract In this paper we compute the distributions of various markings on smooth cubic surfaces defined over the finite field Fq, for example the distribution of pairs of points, ‘tritangents’ or ‘double sixes’. We also compute the (rational) cohomology of certain associated bundles and covers over complex numbers. 1 Introduction The classical Cayley–Salmon theorem implies that each smooth cubic surface over an algebraically closed field contains exactly 27 lines (see Section 2 for detailed definitions). In contrast, for a surface over a finite field Fq, all 27 lines are defined over Fq but not necessarily over Fq itself. In other words, the action of the Frobenius Frobq permutes the 27 lines and only fixes a (possibly empty) subset of them. It is also classical that the group of all such permutations, which can be identified with the Galois group of an appropriate extension or cover, is isomorphic to the Weyl group W(E6) of type E6. This permutation of the 27 lines governs much of the arithmetic of the surface S: evidently the n pattern of lines defined over Fq and, less obviously, the number of Fq points on S (or UConf S etc). Work of Bergvall and Gounelas [BG19] allows us to compute the number of cubic surfaces over Fq where Frobq induces a given permutation, or rather a permutation in a given conjugacy class of W(E6). The results in this paper can be thought of as a combinatorial (Theorem 1.1) or representation-theoretic (Theorem 2.3) reinterpretation of their computation.