Characters of Relative P'-Degree Over Normal Subgroups
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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 ............................ -
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. -
Finite Groups Whose Element Orders Are Consecutive Integers
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 143, 388X)0 (1991) Finite Groups Whose Element Orders Are Consecutive Integers ROLF BRANDL Mathematisches Institui, Am Hubland 12, D-W-8700 Wiirzberg, Germany AND SHI WUJIE Mathematics Department, Southwest-China Teachers University, Chongqing, China Communicated by G. Glauberman Received February 28, 1986 DEDICATED TO PROFESSORB. H. NEUMANN ON HIS 80~~ BIRTHDAY There are various characterizations of groups by conditions on the orders of its elements. For example, in [ 8, 1] the finite groups all of whose elements have prime power order have been classified. In [2, 16-181 some simple groups have been characterized by conditions on the orders of its elements and B. H. Neumann [12] has determined all groups whose elements have orders 1, 2, and 3. The latter groups are OC3 groups in the sense of the following. DEFINITION. Let n be a positive integer. Then a group G is an OC, group if every element of G has order <n and for each m <n there exists an element of G having order m. In this paper we give a complete classification of finite OC, groups. The notation is standard; see [S, 111. In addition, for a prime q, a Sylow q-sub- group of the group G is denoted by P,. Moreover Z{ stands for the direct product of j copies of Zj and G = [N] Q denotes the split extension of a normal subgroup N of G by a complement Q. -
2020 Ural Workshop on Group Theory and Combinatorics
Institute of Natural Sciences and Mathematics of the Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center 2020 Ural Workshop on Group Theory and Combinatorics Yekaterinburg – Online, Russia, August 24-30, 2020 Abstracts Yekaterinburg 2020 2020 Ural Workshop on Group Theory and Combinatorics: Abstracts of 2020 Ural Workshop on Group Theory and Combinatorics. Yekaterinburg: N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, 2020. DOI Editor in Chief Natalia Maslova Editors Vladislav Kabanov Anatoly Kondrat’ev Managing Editors Nikolai Minigulov Kristina Ilenko Booklet Cover Desiner Natalia Maslova c Institute of Natural Sciences and Mathematics of Ural Federal University named after the first President of Russia B.N.Yeltsin N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences The Ural Mathematical Center, 2020 2020 Ural Workshop on Group Theory and Combinatorics Conents Contents Conference program 5 Plenary Talks 8 Bailey R. A., Latin cubes . 9 Cameron P. J., From de Bruijn graphs to automorphisms of the shift . 10 Gorshkov I. B., On Thompson’s conjecture for finite simple groups . 11 Ito T., The Weisfeiler–Leman stabilization revisited from the viewpoint of Terwilliger algebras . 12 Ivanov A. A., Densely embedded subgraphs in locally projective graphs . 13 Kabanov V. V., On strongly Deza graphs . 14 Khachay M. Yu., Efficient approximation of vehicle routing problems in metrics of a fixed doubling dimension . -
On Finite Groups Acting on Acyclic Low-Dimensional Manifolds
On finite groups acting on acyclic low-dimensional manifolds Alessandra Guazzi*, Mattia Mecchia** and Bruno Zimmermann** *SISSA Via Bonomea 256 34136 Trieste, Italy **Universit`adegli Studi di Trieste Dipartimento di Matematica e Informatica 34100 Trieste, Italy Abstract. We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main re- sult states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A5 and A6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds. 1. Introduction All finite group actions considered in the present paper will be faithful and smooth (or locally linear). arXiv:0808.0999v3 [math.GT] 7 Jun 2010 By the recent geometrization of finite group actions on 3-manifolds, every finite group action on the 3-sphere is conjugate to an orthogonal action; in particular, the finite groups which occur are exactly the well-known finite subgroups of the orthogonal group O(4). Finite groups acting on arbitrary homology 3-spheres are considered in [MeZ1] and [Z]; here some other finite groups occur and the situation is still not completely understood (see [Z] and section 7). -
Maximal Subgroups of Finite Groups
MAXIMAL SUBGROUPS OF FINITE GROUPS By LINDSEY-KAY LAUDERDALE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014 c 2014 Lindsey-Kay Lauderdale ⃝ To Mom, Dad and Lisa ACKNOWLEDGMENTS I would like to thank my advisor Dr. Alexandre Turull, who first sparked my interest in group theory. His endless patience and support were invaluable to the completion of this dissertation. I am also grateful for the support, questions and suggestions of my committee members Dr. Kevin Keating, Dr. David Mazyck, Dr. Paul Robinson and Dr. Peter Sin. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 LIST OF TABLES ..................................... 6 ABSTRACT ........................................ 7 CHAPTER 1 DISSERTATION OUTLINE ............................. 9 2 A BRIEF BACKGROUND OF FINITE GROUP THEORY ........... 11 2.1 Introduction ................................... 11 2.2 Maximal Subgroups ............................... 14 2.3 Notation and Some Definitions ......................... 17 2.4 On the Number of Maximal Subgroups .................... 18 3 UPPER BOUNDS FOR THE NUMBER OF MAXIMAL SUBGROUPS .... 20 3.1 Introduction ................................... 20 3.2 Upper Bounds Proven by Others ....................... 21 3.3 Preliminary Results ............................... 22 3.4 Proof of Main Theorem ............................. 23 4 LOWER BOUNDS FOR THE NUMBER OF MAXIMAL SUBGROUPS ... -
E8 + E8 Heterotic String Theory in Vedic Physics
E8 + E8 Heterotic String Theory in Vedic Physics By John Frederic Sweeney Abstract S.M. Phillips has articulated a fairly good model of the E8×E8 heterotic superstring, yet nevertheless has missed a few key aspects. This paper informs his model from the perspective of Vedic Nuclear Physics, as derived from the Rig Veda and two of the Upanishads. In addition, the author hypothesizes an extension of the Exceptional Lie Algebra Series beyond E8 to another 12 places or more. 1 Table of Contents Introduction 3 Wikipedia 5 S.M. Phillips model 12 H series of Hypercircles 14 Conclusion 18 Bibliography 21 2 Introduction S.M. Phillips has done a great sleuthing job in exploring the Jewish Cabala along with the works of Basant and Leadbetter to formulate a model of the Exceptional Lie Algebra E8 to represent nuclear physics that comes near to the Super String model. The purpose of this paper is to offer minor corrections from the perspective of the science encoded in the Rig Veda and in a few of the Upanishads, to render a complete and perfect model. The reader might ask how the Jewish Cabala might offer insight into nuclear physics, since the Cabala is generally thought to date from Medieval Spain. The simple fact is that the Cabala does not represent medieval Spanish thought, it is a product of a much older and advanced society – Remotely Ancient Egypt from 15,000 years ago, before the last major flooding of the Earth and the Sphinx. The Jewish people may very well have left Ancient Egypt in the Exodus, led by Moses. -
On Finite Groups Acting on Homology 4-Spheres and Finite Subgroups Of
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 741–747 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On finite groups acting on homology 4-spheres and finite subgroups of SO(5) ∗ Mattia Mecchia , Bruno Zimmermann Università degli Studi di Trieste, Dipartimento di Matematica e Informatica, 34127 Trieste, Italy article info abstract Article history: We show that a finite group which admits a faithful, smooth, orientation-preserving action Received 30 March 2010 on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of Received in revised form 13 January 2011 the orthogonal group SO(5), by explicitly determining the various groups which can occur Accepted 14 January 2011 (up to an indeterminacy of index two in the case of solvable groups). As a consequence we obtain also a characterization of the finite groups which are isomorphic to subgroups Keywords: Homology 4-sphere of the orthogonal groups SO(5) and O(5). Finite group action © 2011 Elsevier B.V. All rights reserved. Finite simple group 1. Introduction We are interested in the class of finite groups which admit an orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere S4, and to compare it with the class of finite subgroups of the orthogonal group SO(5) acting linearly on S4. As a general hypothesis, all actions in the present paper will be faithful, orientation-preserving and smooth (or locally linear). -
On the Number of Maximal Subgroups in a Finite Group
J. Group Theory 18 (2015), 535–551 DOI 10.1515/jgth-2015-0001 © de Gruyter 2015 On the number of maximal subgroups in a finite group L.-K. Lauderdale Communicated by Nigel Boston Abstract. For a finite group G, let m.G/ denote the set of maximal subgroups of G and .G/ denote the set of primes which divide G . In [4], it is proven that m.G/ .G/ when G is cyclic and if G is noncyclic m.G/j j .G/ p, wherej p .G/j D j is thej smallest prime that divides G . In thisj paper,j we j producej C two new lower2 bounds for m.G/ , both of which considerj j all of the primes in .G/. j j 1 Introduction For any finite group G, let .G/ denote the set of primes which divide G and let j j m.G/ denote the set of maximal subgroups of G. Lower bounds for the number of maximal subgroups have previously been investigated. In [4], three lower bounds are proven. In general, m.G/ .G/ and equality holds if and only if G is j j j j cyclic. Under further assumptions on the structure of G, this lower bound was improved. If G is noncyclic, then m.G/ .G/ p, where p .G/ is the j j j j C 2 smallest prime that divides G . Additionally, if G has a noncyclic Sylow subgroup j j and q .G/ is the smallest prime such that Q Syl .G/ is noncyclic, then it is 2 2 q proven that m.G/ .G/ q: (1.1) j j j j C Observe that given any set of distinct primes q1; q2; : : : ; qn where q1 is the ¹ º smallest, there exists a finite group G with .G/ q1; q2; : : : ; qn and m.G/ .G/ q1: D ¹ º j j D j j C For example, G Zq1 Zq1q2 qn achieves the lower bound in (1.1). -
ALL P-LOCAL FINITE GROUPS of RANK TWO for ODD PRIME P 1
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 4, April 2007, Pages 1725–1764 S 0002-9947(06)04367-4 Article electronically published on November 22, 2006 ALL p-LOCAL FINITE GROUPS OF RANK TWO FOR ODD PRIME p ANTONIO D´IAZ, ALBERT RUIZ, AND ANTONIO VIRUEL Abstract. In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible F- radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corre- sponds to an exotic p-local finite group, getting some new examples of these for p =3. 1. Introduction When studying the p-local homotopy theory of classifying spaces of finite groups, Broto-Levi-Oliver [11] introduced the concept of p-local finite group as a p-local analogue of the classical concept of finite group. These purely algebraic objects, whose basic properties are reviewed in Section 2, are a generalization of the classical theory of finite groups. So every finite group gives rise to a p-local finite group, although there exist exotic p-local finite groups which are not associated to any finite group, as can be read in [11, Sect. 9], [30], [36] or Theorem 5.10 below. Besides its own interest, the systematic study of possible p-local finite groups, i.e. possible p-local structures, is meaningful when working in other research areas such as transformation groups (e.g. -
Maximal Subgroups of Sporadic Groups Was Pe- Tra (Beth) Holmes, Whose Phd Thesis on ‘Computing in the Monster’ Dates from 2002
MAXIMAL SUBGROUPS OF SPORADIC GROUPS ROBERT A. WILSON Abstract. A systematic study of maximal subgroups of the sporadic simple groups began in the 1960s. The work is now almost complete, only a few cases in the Monster remaining outstanding. We give a survey of results obtained, and methods used, over the past 50 years, for the classification of maximal subgroups of sporadic simple groups, and their automorphism groups. 1. Introduction The subtitle of the ‘Atlas of Finite Groups’ [7] is ‘Maximal Subgroups and Or- dinary Characters for Simple Groups’. These two aspects of the study of finite simple groups remain at the forefront of research today. The Atlas was dedicated to collecting facts, not to providing proofs. It contains an extensive bibliography, but not citations at the point of use, making it difficult for the casual reader to track down proofs. In the ensuing 30 years, moreover, the landscape has changed dramatically, both with the appearance of new proofs in the literature, and with the ability of modern computer algebra systems to recompute much of the data in the twinkling of an eye. As far as maximal subgroups are concerned, shortly before the publication of the Atlas it became clear that the maximal subgroup project should be extended to almost simple groups. The reason for this is that it is not possible to deduce the maximal subgroups of an almost simple group directly from the maximal subgroups of the corresponding simple group. This was made clear by the examples described in [49], especially perhaps the maximal subgroup S5 of M12:2, which is neither the normalizer of a maximal subgroup of M12, nor the normalizer of the intersection of two non-conjugate maximal subgroups of M12.