A Brief History of the Classification of the Finite Simple Groups
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Prof. Dr. Eric Jespers Science Faculty Mathematics Department Bachelor Paper II
Prof. Dr. Eric Jespers Science faculty Mathematics department Bachelor paper II 1 Voorwoord Dit is mijn tweede paper als eindproject van de bachelor in de wiskunde aan de Vrije Universiteit Brussel. In dit werk bestuderen wij eindige groepen G die minimaal niet nilpotent zijn in de volgende betekenis, elke echt deelgroep van G is nilpotent maar G zelf is dit niet. W. R. Scott bewees in [?] dat zulke groepen oplosbaar zijn en een product zijn van twee deelgroepen P en Q, waarbij P een cyclische Sylow p-deelgroep is en Q een normale Sylow q-deelgroep is; met p en q verschillende priemgetallen. Het hoofddoel van dit werk is om een volledig en gedetailleerd bewijs te geven. Als toepassing bestuderen wij eindige groepen die minimaal niet Abels zijn. Dit project is verwezenlijkt tijdens mijn Erasmusstudies aan de Universiteit van Granada en werd via teleclassing verdedigd aan de Universiteit van Murcia, waar mijn mijn pro- motor op sabbatical verbleef. Om lokale wiskundigen de kans te geven mijn verdediging bij te wonen is dit project in het Engles geschreven. Contents 1 Introduction This is my second paper to obtain the Bachelor of Mathematics at the University of Brussels. The subject are finite groups G that are minimal not nilpotent in the following meaning. Each proper subgroup of G is nilpotent but G itself is not. W.R. Scott proved in [?] that those groups are solvable and a product of two subgroups P and Q, with P a cyclic Sylow p-subgroup and G a normal Sylow q-subgroup, where p and q are distinct primes. -
Simple Groups, Permutation Groups, and Probability
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 12, Number 2, April 1999, Pages 497{520 S 0894-0347(99)00288-X SIMPLE GROUPS, PERMUTATION GROUPS, AND PROBABILITY MARTIN W. LIEBECK AND ANER SHALEV 1. Introduction In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability, or counting, is not entirely anticipated by the nature of the problem (see [LiSh2], [GSSh]). In this paper we study a variety of problems in finite simple groups and fi- nite permutation groups using a unified method, which often involves probabilistic arguments. We obtain new bounds on the minimal degrees of primitive actions of classical groups, and prove the Cameron-Kantor conjecture that almost sim- ple primitive groups have a base of bounded size, apart from various subset or subspace actions of alternating and classical groups. We use the minimal degree result to derive applications in two areas: the first is a substantial step towards the Guralnick-Thompson genus conjecture, that for a given genus g, only finitely many non-alternating simple groups can appear as a composition factor of a group of genus g (see below for definitions); and the second concerns random generation of classical groups. Our proofs are largely based on a technical result concerning the size of the intersection of a maximal subgroup of a classical group with a conjugacy class of elements of prime order. -
Subgroups of Division Rings in Characteristic Zero Are Characterized
Subgroups of Division Rings Mark Lewis Murray Schacher June 27, 2018 Abstract We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders. MSC(2010): Primary: 16A39, 12E15; Secondary: 16U60 1 Introduction Let F be a field, and fix a and b to be non-0 elements of F . The symbol algebra A =(a, b) is the 4-dimensional algebra over F generated by elements i and j that satisfy the relations: i2 = a, j2 = b, ij = ji. (1) − One usually sets k = ij, which leads to the additional circular relations ij = k = ji, jk = i = kj, ki = j = ki. (2) − − − arXiv:1806.09654v1 [math.RA] 25 Jun 2018 The set 1, i, j, k forms a basis for A over F . It is not difficult to see that A is a central{ simple} algebra over F , and using Wedderburn’s theorem, we see that A is either a 4-dimensional division ring or the ring M2(F ) of2 2 matrices over F . It is known that A is split (i.e. is the ring of 2 2 matrices× over F ) if and only if b is a norm from the field F (√a). Note that×b is a norm over F (√a) if and only if there exist elements x, y F so that b = x2 y2a. This condition is symmetric in a and b. ∈ − More generally, we have the following isomorphism of algebras: (a, b) ∼= (a, ub) (3) 1 where u = x2 y2a is a norm from F (√a); see [4] or [6]. -
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. -
Determining Group Structure from the Sets of Character Degrees
DETERMINING GROUP STRUCTURE FROM THE SETS OF CHARACTER DEGREES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Kamal Aziziheris May 2011 Dissertation written by Kamal Aziziheris B.S., University Of Tabriz, 1999 M.A., University of Tehran, 2001 Ph.D., Kent State University, 2011 Approved by M. L. Lewis, Chair, Doctoral Dissertation Committee Stephen M. Gagola, Jr., Member, Doctoral Dissertation Committee Donald White, Member, Doctoral Dissertation Committee Brett D. Ellman, Member, Doctoral Dissertation Committee Anne Reynolds, Member, Doctoral Dissertation Committee Accepted by Andrew Tonge, Chair, Department of Mathematical Sciences Timothy S. Moerland, Dean, College of Arts and Sciences ii . To the Loving Memory of My Father, Mohammad Aziziheris iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . v INTRODUCTION . 1 1 BACKGROUND RESULTS AND FACTS . 12 2 DIRECT PRODUCTS WHEN cd(Oπ(G)) = f1; n; mg . 22 3 OBTAINING THE CHARACTER DEGREES OF Oπ(G) . 42 4 PROOFS OF THEOREMS D AND E . 53 5 PROOFS OF THEOREMS F AND G . 62 6 EXAMPLES . 70 BIBLIOGRAPHY . 76 iv ACKNOWLEDGEMENTS This dissertation is due to many whom I owe a huge debt of gratitude. I would especially like to thank the following individuals for their support, encouragement, and inspiration along this long and often difficult journey. First and foremost, I offer thanks to my advisor Prof. Mark Lewis. This dissertation would not have been possible without your countless hours of advice and support. Thank you for modeling the actions and behaviors of an accomplished mathematician, an excellent teacher, and a remarkable person. -
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. -
Galois-Equivariant Mckay Bijections for Primes Dividing $ Q-1$
GALOIS-EQUIVARIANT MCKAY BIJECTIONS FOR PRIMES DIVIDING q 1 − A. A. SCHAEFFER FRY Abstract. We prove that for most groups of Lie type, the bijections used by Malle and Sp¨ath in the proof of Isaacs–Malle–Navarro’s inductive McKay conditions for the prime 2 and odd primes dividing q − 1 are also equivariant with respect to certain Galois automorphisms. In particular, this shows that these bijections are candidates for proving Navarro–Sp¨ath–Vallejo’s recently-posited inductive Galois–McKay conditions. Mathematics Classification Number: 20C15, 20C33 Keywords: local-global conjectures, characters, McKay conjecture, Galois-McKay, finite simple groups, Lie type 1. Introduction One of the main sets of questions of interest in the representation theory of finite groups falls under the umbrella of the so-called local-global conjectures, which relate the character theory of a finite group G to that of certain local subgroups. Another rich topic in the area is the problem of determining how certain other groups (for example, the group of automorphisms Aut(G) of G, or the Galois group Gal(Q/Q)) act on the set of irreducible characters of G and the related question of determining the fields of values of characters of G. This paper concerns the McKay–Navarro conjecture, which incorporates both of these main problems. For a finite group G, the field Q(e2πi/|G|), obtained by adjoining the G th roots of unity in some | | algebraic closure Q of Q, is a splitting field for G. The group G := Gal(Q(e2πi/|G|)/Q) acts naturally on the set of irreducible ordinary characters, Irr(G), of G via χσ(g) := χ(g)σ for χ Irr(G), g G, σ G . -
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, . -
Discrete and Profinite Groups Acting on Regular Rooted Trees
Discrete and Profinite Groups Acting on Regular Rooted Trees Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen vorgelegt von Olivier Siegenthaler aus Lausanne G¨ottingen, den 31. August 2009 Referent: Prof. Dr. Laurent Bartholdi Korreferent: Prof. Dr. Thomas Schick Tag der m¨undlichen Pr¨ufung: den 28. September 2009 Contents Introduction 1 1 Foundations 5 1.1 Definition of Aut X∗ and ...................... 5 A∗ 1.2 ZariskiTopology ............................ 7 1.3 Actions of X∗ .............................. 8 1.4 Self-SimilarityandBranching . 9 1.5 Decompositions and Generators of Aut X∗ and ......... 11 A∗ 1.6 The Permutation Modules k X and k X ............ 13 { } {{ }} 1.7 Subgroups of Aut X∗ .......................... 14 1.8 RegularBranchGroups . 16 1.9 Self-Similarity and Branching Simultaneously . 18 1.10Questions ................................ 19 ∗ 2 The Special Case Autp X 23 2.1 Definition ................................ 23 ∗ 2.2 Subgroups of Autp X ......................... 24 2.3 NiceGeneratingSets. 26 2.4 Uniseriality ............................... 28 2.5 Signature and Maximal Subgroups . 30 2.6 Torsion-FreeGroups . 31 2.7 Some Specific Classes of Automorphisms . 32 2.8 TorsionGroups ............................. 34 3 Wreath Product of Affine Group Schemes 37 3.1 AffineSchemes ............................. 38 3.2 ExponentialObjects . 40 3.3 Some Hopf Algebra Constructions . 43 3.4 Group Schemes Corresponding to Aut X∗ .............. 45 3.5 Iterated Wreath Product of the Frobenius Kernel . 46 i Contents 4 Central Series and Automorphism Towers 49 4.1 Notation................................. 49 4.2 CentralSeries.............................. 51 4.3 Automorphism and Normalizer Towers . 54 5 Hausdorff Dimension 57 5.1 Definition ................................ 57 5.2 Layers .................................. 58 5.3 ComputingDimensions . -
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. -
Generalized Quaternions
GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8. -
Abstract Quotients of Profinite Groups, After Nikolov and Segal
ABSTRACT QUOTIENTS OF PROFINITE GROUPS, AFTER NIKOLOV AND SEGAL BENJAMIN KLOPSCH Abstract. In this expanded account of a talk given at the Oberwolfach Ar- beitsgemeinschaft “Totally Disconnected Groups”, October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is [17]. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject. 1. Introduction §1.1. Many concepts and techniques in the theory of finite groups depend in- trinsically on the assumption that the groups considered are a priori finite. The theoretical framework based on such methods has led to marvellous achievements, including – as a particular highlight – the classification of all finite simple groups. Notwithstanding, the same methods are only of limited use in the study of infinite groups: it remains mysterious how one could possibly pin down the structure of a general infinite group in a systematic way. Significantly more can be said if such a group comes equipped with additional information, such as a structure-preserving action on a notable geometric object. A coherent approach to studying restricted classes of infinite groups is found by imposing suitable ‘finiteness conditions’, i.e., conditions that generalise the notion of being finite but are significantly more flexible, such as the group being finitely generated or compact with respect to a natural topology. One rather fruitful theme, fusing methods from finite and infinite group theory, consists in studying the interplay between an infinite group Γ and the collection of all its finite quotients.