DOCSLIB.ORG

Galois Groups of Trinomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Galois Groups of Trinomials Journal of Algebra 222, 561–573 (1999) Article ID jabr.1999.8033, available online at http://www.idealibrary.com on Galois Groups of Trinomials S. D. Cohen Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland E-mail: [email protected] View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector A. Movahhedi and A. Salinier URA 1586 CNRS, Facult´e des Sciences, 123 Avenue Albert Thomas, 87060 Limoges Cedex, France Communicated by Walter Feit Received May 7, 1999 Galois groups of irreducible trinomials Xn C aXs C b 2 X are investigated assuming the classification of finite simple groups. We show that under some simple yet general hypotheses bearing on the integers n; s; a and b only very specific groups can occur. For instance, if the two integers nb and asn − s are coprime and if s is a prime number, then already the Galois group of f X is either the alternating group An or the symmetric group Sn. This significantly extends work of Osada. © 1999 Academic Press 1. INTRODUCTION Let f X=Xn C aXs C b be an irreducible trinomial with integral coef- ficients, where n and s are co-prime. Although, of course, we know that, in a sense, almost all such trinomials must have the symmetric group Sn as Ga- lois group, it is valuable to identify simple and explicit, yet general, criteria on the coefficients a, b under which this is the case. Examples of such cri- teria occur in the work of Osada [11, 12]: we greatly extend these to cover situations which, by any reckoning, are more delicate (because non-tame ramification is involved). Thus, whereas in a preceding paper [3], we estab- lished the double transitivity of the Galois group of f X over under 561 0021-8693/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved. 562 cohen, movahhedi, and salinier some general circumstances, now we use the classification of finite simple groups and other special considerations to specify in most cases which dou- bly transitive groups are liable to be realized in this way; there are very few possibilities. Indeed, as we remarked in [3], generally not many trinomials for which the alternating group An 6⊆ Gf are known; perhaps the results of the paper may help to confine the search for such examples to a smaller area. Denote by α xD α1;α2;:::;αn the different roots of f X in an algebraic closure of .LetK xD α be the field obtained by adjoining the root α to the field , and L xD α,α2, :::,αn be the normal closure of K over . We consider the Galois group G of L over as a transitive group of permutations of the roots of f X. For a prime p, vpb denotes the p-adic valuation of an integer b, i.e., the largest integer k ≥ 0 such that pk divides b. The first result extends Theorem 1.1 of [3]. Theorem 1.1. Let f X=Xn C aXs C b be an irreducible trinomial with integral coefficients, where n; as=an − s;b=1. Suppose there exists a prime divisor p of b such that s; vpb D 1. Suppose the Galois group G of f X over does not contain the alternating group An. Then s>n=2. Indeed, if s 6D n − 1, the following hold. (1) If n=2 <s<n− 2, we must have n =qd − 1=q − 1 ;sD qd − qe=q − 1, where q is an odd prime power, d is an even integer, e an integer coprime to d with 2 <e<d, the prime p is a divisor of s and PSLd; q⊆G ⊆ P0Ld; q: (2) If s D n − 2, only the following two possibilities can occur. (2a) There is an odd integer m such that n D 2m C 1 and G D P0L2; 2m. (2b) n D 11, p D 3 and G is the Mathieu group M11. Under the conditions of Theorem 1.1, the double transitivity of G was established in Theorem 1.1 of [3]. Of course, doubly transitive groups have been classified, but the significance of Theorem 1.1 is that, provided s< n − 1, most of these groups cannot arise. When s D n − 1 no progress is made in this paper on the resolution of the problem of which doubly transitive groups may occur. In the case in which additionally b; s=1, the following consequence of Theorem 1.1 is an improvement (in a sense, a completion) of [12, Theo- rem 1] by the omission of any condition on the discriminant of f X in its statement. galois groups of trinomials 563 Theorem 1.2. Let f X=Xn C aXs C b be an irreducible trinomial with integral coefficients, where s 6D n − 1 and nb; asn − s D 1: Then the Galois group G of f X over contains the alternating group An in each of the following two cases. (i) There exists a prime divisor p of b such that s; vpb D 1; (ii) s is a prime number. Of course, under the conditions of Theorem 1.2, G D An if and only if the discriminant of f X is a square. A further result extends Theorem 1.2 of [3]. Theorem 1.3. Let f X=Xn C aXs C b be an irreducible trinomial with integral coefficients, where s>1 and n; s=1. Suppose that there exists a prime divisor p of b, but not of a, such that (i) n D s C pt ;t>0, (ii) vpf −a D 1, (iii) s; vpb D 1. Suppose also that the Galois group G of f X over does not contain the alternating group An. Then only the following possibilities can occur. 1. s D 2, t D 1, n D p C 2, where p D 2m − 1 is a Mersenne prime, and the Galois group G is either PGL2; 2m or P0L2; 2m. 2. s D 2, n D 11, p D 3 and the Galois group G is the Mathieu group M11. 3. s D 3, n D 11, p D 2 and the Galois group G is the Mathieu group M11. 4. s D 1 C q C···Cqd−2, n D 1 C q C···Cqd−1, and the Galois group G lies between PSLd; q and P0Ld; q, where (4a) q D 2 and d D 3 or 4,ifp D 2; (4b) q D pr and d is an even integer ≥ 4,ifp is odd. We stress that the validity of Theorem 1.3 has (deliberately) been stated under rather general conditions. For instance, it is not supposed that a and nb are coprime. We are aware that, by relaxing these conditions in various ways, some of the possibilities allowed by the theorem may be discounted. For example, if a and b are assumed to be coprime, then we know that the possibilities described in (4a) (namely, with p D q D 2 and d D 3or 4) cannot arise. Whether any of the possibilities (1), (2a), (2b) of Theorem 1.1 and (1)–(4) of Theorem 1.3 can be realized by rational trinomials is an open question about which we lack evidence either way. 564 cohen, movahhedi, and salinier 2. SOME FACTS FROM GROUP THEORY Here we gather some facts on permutation groups that we shall use re- peatedly in the subsequent sections. 2.1. Jordan Groups Let G be a primitive permutation group of degree n acting on a set X. For 1 <r<n− 1, we say that G is an n; r-Jordan group if the following conditions hold: (1) G is not r C 1-fold transitive; (2) there exists a subset Y (“a Jordan complement”) of X of car- dinality r whose pointwise stabilizer acts transitively on the set X n Y of remaining points. Jordan groups have been classified [10, p. 276](see also [4, Section 7.4]). They are divided into three categories. First there are the affine groups, which are qd;qe-Jordan groups, where q is a prime power and d and e are integers such that d>e>0. The second category comprises the groups G acting on a projective space PGd−1q of dimension d − 1 over the field q (with q a prime power and d ≥ 3) such that PSLd; q⊆ G ⊆ P0Ld; q: they are qd − 1=q − 1; qe − 1=q − 1-Jordan groups for 1 <e<d. Finally, there are exceptional Jordan groups for n; r in 15; 3; 16; 4; 22; 6; 23; 7; 24; 8. In particular, the exceptional 23; 7-Jordan group is the Mathieu group M23. When n and r are coprime, the only n; r-Jordan groups are the Mathieu d group M23 and the projective groups. It is not hard to see that q − 1=q − 1 and qe − 1=q − 1 are coprime if and only if d and e are coprime. Accordingly, we have the following proposition. Proposition 2.1. Let G be an n; r-Jordan group with n and r coprime. Then one of the following holds. 1. There are two coprime integers d and e with 1 <e<dand a prime power q such that n =qd − 1=q − 1, r =qe − 1=q − 1, and PSLd; q⊆G ⊆ P0Ld; q. 2. n D 23;r D 7 and G D M23.
Load more

Recommended publications
  • An Alternative Existence Proof of the Geometry of Ivanov–Shpectorov for O'nan's Sporadic Group
    Innovations in Incidence Geometry Volume 15 (2017), Pages 73–121 ISSN 1781-6475 An alternative existence proof of the geometry of Ivanov–Shpectorov for O’Nan’s sporadic group Francis Buekenhout Thomas Connor In honor of J. A. Thas’s 70th birthday Abstract We provide an existence proof of the Ivanov–Shpectorov rank 5 diagram geometry together with its boolean lattice of parabolic subgroups and es- tablish the structure of hyperlines. Keywords: incidence geometry, diagram geometry, Buekenhout diagrams, O’Nan’s spo- radic group MSC 2010: 51E24, 20D08, 20B99 1 Introduction We start essentially but not exclusively from: • Leemans [26] giving the complete partially ordered set ΛO′N of conjugacy classes of subgroups of the O’Nan group O′N. This includes 581 classes and provides a structure name common for all subgroups in a given class; • the Ivanov–Shpectorov [24] rank 5 diagram geometry for the group O′N, especially its diagram ∆ as in Figure 1; ′ • the rank 3 diagram geometry ΓCo for O N due to Connor [15]; • detailed data on the diagram geometries for the groups M11 and J1 [13, 6, 27]. 74 F. Buekenhout • T. Connor 4 1 5 0 1 3 P h 1 1 1 2 Figure 1: The diagram ∆IvSh of the geometry ΓIvSh Our results are the following: • we get the Connor geometry ΓCo as a truncation of the Ivanov–Shpectorov geometry ΓIvSh (see Theorem 7.1); • using the paper of Ivanov and Shpectorov [24], we establish the full struc- ture of the boolean lattice LIvSh of their geometry as in Figure 17 (See Section 8); • conversely, within ΛO′N we prove the existence and uniqueness up to fu- ′ sion in Aut(O N) of a boolean lattice isomorphic to LIvSh.
    [Show full text]
  • The Book of Abstracts
    1st Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society Victoria University of Wellington Wellington, New Zealand December 12{15, 2007 American Mathematical Society New Zealand Mathematical Society Contents Timetables viii Plenary Addresses . viii Special Sessions ............................. ix Computability Theory . ix Dynamical Systems and Ergodic Theory . x Dynamics and Control of Systems: Theory and Applications to Biomedicine . xi Geometric Numerical Integration . xiii Group Theory, Actions, and Computation . xiv History and Philosophy of Mathematics . xv Hopf Algebras and Quantum Groups . xvi Infinite Dimensional Groups and Their Actions . xvii Integrability of Continuous and Discrete Evolution Systems . xvii Matroids, Graphs, and Complexity . xviii New Trends in Spectral Analysis and PDE . xix Quantum Topology . xx Special Functions and Orthogonal Polynomials . xx University Mathematics Education . xxii Water-Wave Scattering, Focusing on Wave-Ice Interactions . xxiii General Contributions . xxiv Plenary Addresses 1 Marston Conder . 1 Rod Downey . 1 Michael Freedman . 1 Bruce Kleiner . 2 Gaven Martin . 2 Assaf Naor . 3 Theodore A Slaman . 3 Matt Visser . 4 Computability Theory 5 George Barmpalias . 5 Paul Brodhead . 5 Cristian S Calude . 5 Douglas Cenzer . 6 Chi Tat Chong . 6 Barbara F Csima . 6 QiFeng ................................... 6 Johanna Franklin . 7 Noam Greenberg . 7 Denis R Hirschfeldt . 7 Carl G Jockusch Jr . 8 Bakhadyr Khoussainov . 8 Bj¨ornKjos-Hanssen . 8 Antonio Montalban . 9 Ng, Keng Meng . 9 Andre Nies . 9 i Jan Reimann . 10 Ludwig Staiger . 10 Frank Stephan . 10 Hugh Woodin . 11 Guohua Wu . 11 Dynamical Systems and Ergodic Theory 12 Boris Baeumer . 12 Mathias Beiglb¨ock . 12 Arno Berger . 12 Keith Burns . 13 Dmitry Dolgopyat . 13 Anthony Dooley .
    [Show full text]
  • Bogomolov F., Tschinkel Yu. (Eds.) Geometric Methods in Algebra And
    Progress in Mathematics Volume 235 Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein Geometric Methods in Algebra and Number Theory Fedor Bogomolov Yuri Tschinkel Editors Birkhauser¨ Boston • Basel • Berlin Fedor Bogomolov Yuri Tschinkel New York University Princeton University Department of Mathematics Department of Mathematics Courant Institute of Mathematical Sciences Princeton, NJ 08544 New York, NY 10012 U.S.A. U.S.A. AMS Subject Classifications: 11G18, 11G35, 11G50, 11F85, 14G05, 14G20, 14G35, 14G40, 14L30, 14M15, 14M17, 20G05, 20G35 Library of Congress Cataloging-in-Publication Data Geometric methods in algebra and number theory / Fedor Bogomolov, Yuri Tschinkel, editors. p. cm. – (Progress in mathematics ; v. 235) Includes bibliographical references. ISBN 0-8176-4349-4 (acid-free paper) 1. Algebra. 2. Geometry, Algebraic. 3. Number theory. I. Bogomolov, Fedor, 1946- II. Tschinkel, Yuri. III. Progress in mathematics (Boston, Mass.); v. 235. QA155.G47 2004 512–dc22 2004059470 ISBN 0-8176-4349-4 Printed on acid-free paper. c 2005 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Springer Science+Business Media Inc., Rights and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
    [Show full text]
  • Title: Algebraic Group Representations, and Related Topics a Lecture by Len Scott, Mcconnell/Bernard Professor of Mathemtics, the University of Virginia
    Title: Algebraic group representations, and related topics a lecture by Len Scott, McConnell/Bernard Professor of Mathemtics, The University of Virginia. Abstract: This lecture will survey the theory of algebraic group representations in positive characteristic, with some attention to its historical development, and its relationship to the theory of finite group representations. Other topics of a Lie-theoretic nature will also be discussed in this context, including at least brief mention of characteristic 0 infinite dimensional Lie algebra representations in both the classical and affine cases, quantum groups, perverse sheaves, and rings of differential operators. Much of the focus will be on irreducible representations, but some attention will be given to other classes of indecomposable representations, and there will be some discussion of homological issues, as time permits. CHAPTER VI Linear Algebraic Groups in the 20th Century The interest in linear algebraic groups was revived in the 1940s by C. Chevalley and E. Kolchin. The most salient features of their contributions are outlined in Chapter VII and VIII. Even though they are put there to suit the broader context, I shall as a rule refer to those chapters, rather than repeat their contents. Some of it will be recalled, however, mainly to round out a narrative which will also take into account, more than there, the work of other authors. §1. Linear algebraic groups in characteristic zero. Replicas 1.1. As we saw in Chapter V, §4, Ludwig Maurer thoroughly analyzed the properties of the Lie algebra of a complex linear algebraic group. This was Cheval­ ey's starting point.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Polytopes Derived from Sporadic Simple Groups
    Volume 5, Number 2, Pages 106{118 ISSN 1715-0868 POLYTOPES DERIVED FROM SPORADIC SIMPLE GROUPS MICHAEL I. HARTLEY AND ALEXANDER HULPKE Abstract. In this article, certain of the sporadic simple groups are analysed, and the polytopes having these groups as automorphism groups are characterised. The sporadic groups considered include all with or- der less than 4030387201, that is, all up to and including the order of the Held group. Four of these simple groups yield no polytopes, and the highest ranked polytopes are four rank 5 polytopes each from the Higman-Sims group, and the Mathieu group M24. 1. Introduction The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if 2 0 the Tits group F4(2) is counted also) which these infinite families do not include. These sporadic simple groups range in size from the Mathieu group 53 M11 of order 7920, to the Monster group M of order approximately 8×10 . One key to the study of these groups is identifying some geometric structure on which they act. This provides intuitive insight into the structure of the group. Abstract regular polytopes are combinatorial structures that have their roots deep in geometry, and so potentially also lend themselves to this purpose. Furthermore, if a polytope is found that has a sporadic group as its automorphism group, this gives (in theory) a presentation of the sporadic group over some generating involutions. In [6], a simple algorithm was given to find all polytopes acted on by a given abstract group.
    [Show full text]
  • Experimental Mathematics
    A Complete Bibliography of Publications in Experimental Mathematics Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 30 March 2021 Version 1.28 Title word cross-reference (((X2 − P )2 − Q)2 − R)2 − S2 [Bre08a]. (−2; 3;n) [GK12]. (1; 2) [EF18]. (2; 3; 6) [Sch06]. (4k + 2) [AJK08]. {−1; 0; 1g [SS06]. 0 [Cum04, Har15]. 1 [BR18, BM00, CF95, Cum04, Fin92]. 1=2 [Car03]. 1/π [Gui06a, Gui06b]. 1/π2 [AG12, Gui06b]. 100 [BCVZ02]. 10651 [MR95]. 1019 [AD07]. 1027 [RT16]. 11 [BLP09, BGKT05, Lep93, LM19]. 1132 [KP08]. 12 [GP94, LM19]. 128 [JM14, Elk01]. 15 [BGKT05, DL92, KO95]. 16 [EMRSE14,´ MTC14]. 19 [CHLM97]. 191 [HPP08]. 1 <n<10 [OM08]. 1: 4 [Kra94]. 2 [AT00, AMM08, BLP09, CD09, Elk01, EL14, KdC05, KV17, Kat10b, Man18, Nic18, RW11, Van17]. 23 [DGKMY15]. 24 [CP03]. 2 × 2 [CILO19, LR17]. 3 [ABL17b, AT00, Ban06, BFG+13, BL00, Bur04, But05, CLLM07, CLT06, CGHN00, Der15, DBG19, EM02, FMP04, Fuk11, GHL19, HS15, HIK+16, Kat10b, KY17, LR07, Sha19, Tan09, Ume09, Web97]. 32 [BN97, CH08a]. 3630 [BCVZ02]. 3n + 1 [LSW99]. 3x + 1 [AL95, BM98, Nic18]. 3x + d [BM98]. 4 [Con06, JK01, RT00, SZ04]. 47 [Tak14]. 5 [De 15a, HS07, Kat10b]. 6 [ALT+12, FP92, FP93, FPDH98, HP08]. 6; 6 [Nil03]. 7 [ALT+12, BGKT05]. 1 2 71 [Tak14]. 8 [ALT+12, BN97, Bur07, FJ96].
    [Show full text]
  • The Character Table of a Sharply 5-Transitive Subgroup of the Alternating Group of Degree 12
    International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 www.theoryofgroups.ir Vol. x No. x (201x), pp. xx-xx. www.ui.ac.ir c 201x University of Isfahan THE CHARACTER TABLE OF A SHARPLY 5-TRANSITIVE SUBGROUP OF THE ALTERNATING GROUP OF DEGREE 12 NICK GILL AND SAM HUGHES Communicated by Abstract. We calculate the character table of a sharply 5-transitive subgroup of Alt(12), and of a sharply 4-transitive subgroup of Alt(11). Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations. 1. Introduction Let Alt(n) denote the alternating group on n points. In this paper we will be interested in subgroups of Alt(n) that are sharply k-transitive, for some integer k ≥ 4. Recall that a permutation group G acting on a set Ω is sharply k-transitive if, first, G acts transitively on the set of k-tuples of distinct elements of Ω and, second, the (point-wise) stabilizer of any k-tuple of distinct elements of Ω is trivial. Notice that \1-transitivity" is the same as \transitivity". In this paper we prove the following theorem. Theorem 1.1. (1) If G is a sharply 5-transitive subgroup of Alt(12), then its character table is given by Table 1. (2) If G is a sharply 4-transitive subgroup of Alt(11), then its character table is given by Table 2.
    [Show full text]
  • Finite Simple Groups and Localization
    Finite simple groups and localization Jose´ L. Rodr´ıguez, Jer´ omeˆ Scherer and Jacques Thevenaz´ ∗ Abstract The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup and we apply it in various cases. Iterating this process allows us to connect many simple groups by a sequence of localizations. We prove that all sporadic simple groups (except possibly the Monster) and several groups of Lie type are connected to alternating groups. The question remains open whether or not there are several connected components within the family of finite simple groups. In some cases, we also consider automorphism groups and universal covering groups and we show that a localization of a finite simple group may not be simple. Introduction The concept of localization plays an important role in homotopy theory. The introduction by Bousfield of homotopical localization functors in [2] and more recently its populariza- tion by Farjoun in [7] has led to the study of localization functors in other categories. Special attention has been set on the category of groups Gr, as the effect of a homotopi- cal localization on the fundamental group is often best described by a localization functor L:Gr → Gr. A localization functor is a pair (L,η) consisting of a functor L : Gr → Gr together with a natural transformation η :Id→ L, such that L is idempotent, meaning that the two morphisms ηLG, L(ηG):LG → LLG coincide and are isomorphisms.
    [Show full text]
  • Constructions and Simplicity of the Mathieu Groups
    CONSTRUCTIONS AND SIMPLICITY OF THE MATHIEU GROUPS a thesis submitted to the graduate school of engineering and science of bilkent university in partial fulfillment of the requirements for the degree of master of science in mathematics By Mete Han Karaka¸s August 2020 CONSTRUCTIONS AND SIMPLICITY OF THE MATHIEU GROUPS By Mete Han Karaka¸s August 2020 We certify that we have read this thesis and that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science. Matthew Justin Karcher Gelvin (Advisor) Laurence John Barker Omer¨ K¨u¸c¨uksakallı Approved for the Graduate School of Engineering and Science: Ezhan Kara¸san Director of the Graduate School ii ABSTRACT CONSTRUCTIONS AND SIMPLICITY OF THE MATHIEU GROUPS Mete Han Karaka¸s M.S. in Mathematics Advisor: Matthew Justin Karcher Gelvin August 2020 Of the 26 sporadic finite simple groups, 5 were discovered by E. Mathieu in 1861 and 1873 [1], [2]. These Mathieu groups are the focus of this thesis, where we will prove their simplicity using elementary methods. E. Witt [5] realized a connection between the Mathieu groups and certain combinatorial structures known as Steiner systems. We will follow his construction to define the Mathieu groups as the auto- morphism groups of certain Steiner systems. Much of the work of the thesis lies in the construction of these Steiner systems, which we achieve by using both methods from finite geometry and the theory of Golay codes. Keywords: Mathieu groups, Steiner systems, Golay codes. iii OZET¨ MATHIEU GRUPLARININ OLUS¸TURULMASI VE BASITL_ I_G˘ I_ Mete Han Karaka¸s Matematik, Y¨uksekLisans Tez Danı¸smanı:Matthew Justin Karcher Gelvin A˘gustos2020 26 tane sporadik sonlu basit gruplardan 5 tanesi 1861 ve 1873 yıllarında E.
    [Show full text]
  • Finite Groups Acting on Hyperelliptic 3-Manifolds 3
    FINITE GROUPS ACTING ON HYPERELLIPTIC 3-MANIFOLDS MATTIA MECCHIA Abstract. We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to S3. Such involutions are called hyperelliptic as the man- ifolds admitting such an action. We consider finite groups acting on 3-manifolds and con- taining hyperelliptic involutions whose fixed-point set has r > 2 components. In particular we prove that a simple group containing such an involution is isomorphic to PSL(2, q) for some odd prime power q, or to one of four other small simple groups. 1. Introduction In the present paper, we consider hyperelliptic 3-manifolds; these are 3-manifolds which admit the action of a hyperelliptic involution, i.e. an involution with quotient space home- omorphic to S3. In this paper 3-manifolds are all smooth, closed and orientable, and the actions of groups on 3-manifolds are smooth and orientation-preserving. The analogous concept in dimension 2 is a classical research theme and many papers about hyperelliptic Riemann surfaces can be found in the literature. In particular the hy- perelliptic involution of a Riemann surface of genus at least two is unique and central in the automorphism group of the hyperelliptic Riemann surface; as a consequence, the class of automorphism groups of hyperelliptic Riemann surfaces is very restricted (liftings of finite groups acting on S2). In dimension 3 the presence of a hyperelliptic involution is equivalent to the fact that the 3-manifold is the 2-fold cover of S3 branched along a link (in brief the 2-fold branched cover of a link).
    [Show full text]