Modular Representations of the Hall-Janko Group
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ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆. -
ON the INTERSECTION NUMBER of FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler
University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-14-2019 ON THE INTERSECTION NUMBER OF FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Algebra Commons, and the Discrete Mathematics and Combinatorics Commons Recommended Citation Bautista Serrano, Humberto, "ON THE INTERSECTION NUMBER OF FINITE GROUPS" (2019). Math Theses. Paper 9. http://hdl.handle.net/10950/1332 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. ON THE INTERSECTION NUMBER OF FINITE GROUPS by HUMBERTO BAUTISTA SERRANO A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Kassie Archer, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler April 2019 c Copyright by Humberto Bautista Serrano 2019 All rights reserved Acknowledgments Foremost I would like to express my gratitude to my two excellent advisors, Dr. Kassie Archer at UT Tyler and Dr. Lindsey-Kay Lauderdale at Towson University. This thesis would never have been possible without their support, encouragement, and patience. I will always be thankful to them for introducing me to research in mathematics. I would also like to thank the reviewers, Dr. Scott LaLonde and Dr. David Milan for pointing to several mistakes and omissions and enormously improving the final version of this thesis. -
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. -
(Hereditarily) Just Infinite Property in Profinite Groups
Inverse system characterizations of the (hereditarily) just infinite property in profinite groups Colin D. Reid October 6, 2018 Abstract We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are ‘universal’ in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. This is a corrected and revised version of [8]. 1 Introduction Notation. In this paper, all groups will be profinite groups, all homomorphisms are required to be continuous, and all subgroups are required to be closed; in particular, all references to commutator subgroups are understood to mean the closures of the corresponding abstractly defined subgroups. For an inverse system Λ= {(Gn)n>0, ρn : Gn+1 ։ Gn} of finite groups, we require all the homomorphisms ρn to be surjective. A subscript o will be used to indicate open inclusion, for instance A ≤o B means that A is an open subgroup of B. We use ‘pronilpotent’ and ‘prosoluble’ to mean a group that is the inverse limit of finite nilpotent groups or finite soluble groups respectively, and ‘G-invariant subgroup of H’ to mean a subgroup of H normalized by G. A profinite group G is just infinite if it is infinite, and every nontrivial normal subgroup of G is of finite index; it is hereditarily just infinite if in addition every arXiv:1708.08301v1 [math.GR] 28 Aug 2017 open subgroup of G is just infinite. At first sight the just infinite property is a qualitative one, like that of simplicity: either a group has nontrivial normal subgroups of infinite index, or it does not. -
Verification of Dade's Conjecture for Janko Group J3
JOURNAL OF ALGEBRA 187, 579]619Ž. 1997 ARTICLE NO. JA966836 Verification of Dade's Conjecture for Janko Group J3 Sonja KotlicaU Department of Mathematics, Uni¨ersity of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801 View metadata, citation and similar papersCommunicated at core.ac.uk by Walter Feit brought to you by CORE provided by Elsevier - Publisher Connector Received July 5, 1996 Inwx 3 Dade made a conjecture expressing the number kBŽ.,d of characters of a given defect d in a given p-block B of a finite group G in terms of the corresponding numbers kbŽ.,d for blocks b of certain p-local subgroups of G. Several different forms of this conjecture are given inwx 5 . Dade claims that the most complicated form of this conjecture, called the ``Inductive Conjecture 5.8'' inwx 5 , will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive conjecture for all covering groups of the third Janko group J3 Žin the notation of the Atlaswx 1. This is one step in the inductive proof of the conjecture for all finite groups. Certain properties of J33simplify our task. The Schur Multiplier of J is cyclic of order 3Ž seewx 1, p. 82. Hence, there are just two covering groups of J33, namely J itself and a central extension 3 ? J33of J by a cyclic group Z of order 3. We treat these two covering groups separately. The outer automorphism group OutŽ.J33of J is cyclic of order 2 Ž seewx 1, p. -
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. -
Orders on Computable Torsion-Free Abelian Groups
Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24 Outline 1 Classical Algebra Background 2 Computing a Basis 3 Computing an Order With A Basis Without A Basis 4 Open Questions Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24 Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = (G : +; 0) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if (8x 2 G)(8n 2 !)[x 6= 0 ^ n 6= 0 =) nx 6= 0] : Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24 Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup ! of Q . Definition The rank of a countable torsion-free abelian group G is the least κ cardinal κ such that G is a subgroup of Q . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24 Example The subgroup H of Q ⊕ Q (viewed as having generators b1 and b2) b1+b2 generated by b1, b2, and 2 b1+b2 So elements of H look like β1b1 + β2b2 + α 2 for β1; β2; α 2 Z. -
Symmetric Generation and Existence of J3:2, the Automorphism Group of the Third Janko Group
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 304 (2006) 256–270 www.elsevier.com/locate/jalgebra Symmetric generation and existence of J3:2, the automorphism group of the third Janko group J.D. Bradley, R.T. Curtis ∗ School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 27 June 2005 Available online 28 February 2006 Communicated by Jan Saxl Abstract We make use of the primitive permutation action of degree 120 of the automorphism group of L2(16), the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J3 and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J3:2, to obtain the order of J3 and to prove its simplicity. A presentation for J3:2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J3:2. © 2006 Elsevier Inc. All rights reserved. 1. Introduction The group J3 was predicted by Janko in a paper in 1968 [10]. He considered a finite simple group with a single class of involutions in which the centraliser of an involution 1+4 was isomorphic to 2 :A5. The following year Higman and McKay, using an observation of Thompson that J3 must contain a subgroup isomorphic to L2(16):2, constructed the group on a computer [9]. -
Arxiv:1509.08090V1 [Math.GR]
THE CLASS MN OF GROUPS IN WHICH ALL MAXIMAL SUBGROUPS ARE NORMAL AGLAIA MYROPOLSKA Abstract. We investigate the class MN of groups with the property that all maximal subgroups are normal. The class MN appeared in the framework of the study of potential counter-examples to the Andrews-Curtis conjecture. In this note we give various structural properties of groups in MN and present examples of groups in MN and not in MN . 1. Introduction The class MN was introduced in [Myr13] as the class of groups with the property that all maximal subgroups are normal. The study of MN was motivated by the analysis of potential counter-examples to the Andrews-Curtis conjecture [AC65]. It was shown in [Myr13] that a finitely generated group G in the class MN satisfies the so-called “generalised Andrews- Curtis conjecture” (see [BLM05] for the precise definition) and thus cannot confirm potential counter-examples to the original conjecture. Apart from its relation to the Andrews-Curtis conjecutre, the study of the class MN can be interesting on its own. Observe that if a group G belongs to MN then all maximal subgroups of G are of finite index. The latter group property has been considered in the literature for different classes of groups. For instance in the linear setting, Margulis and Soifer [MS81] showed that all maximal subgroups of a finitely generated linear group G are of finite index if and only if G is virtually solvable. The above property also was considered for branch groups, however the results in this direction are partial and far from being as general as for linear groups. -
Dihedral Groups Ii
DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of Dn, including the normal subgroups. We will also introduce an infinite group that resembles the dihedral groups and has all of them as quotient groups. 1. Abstract characterization of Dn −1 −1 The group Dn has two generators r and s with orders n and 2 such that srs = r . We will show every group with a pair of generators having properties similar to r and s admits a homomorphism onto it from Dn, and is isomorphic to Dn if it has the same size as Dn. Theorem 1.1. Let G be generated by elements x and y where xn = 1 for some n ≥ 3, 2 −1 −1 y = 1, and yxy = x . There is a surjective homomorphism Dn ! G, and if G has order 2n then this homomorphism is an isomorphism. The hypotheses xn = 1 and y2 = 1 do not mean x has order n and y has order 2, but only that their orders divide n and divide 2. For instance, the trivial group has the form hx; yi where xn = 1, y2 = 1, and yxy−1 = x−1 (take x and y to be the identity). Proof. The equation yxy−1 = x−1 implies yxjy−1 = x−j for all j 2 Z (raise both sides to the jth power). Since y2 = 1, we have for all k 2 Z k ykxjy−k = x(−1) j by considering even and odd k separately. Thus k (1.1) ykxj = x(−1) jyk: This shows each product of x's and y's can have all the x's brought to the left and all the y's brought to the right. -
Math 602 Assignment 1, Fall 2020 Part A
Math 602 Assignment 1, Fall 2020 Part A. From Gallier{Shatz: Problems 1, 2, 6, 9. Part B. Definition Suppose a group G operates transitively on the left of a set X. (i) We say that the action (G; X) is imprimitive if there exists a non-trivial partition Q of X which is stable under G. Here \non-trivial" means that Q 6= fXg and Q 6= ffxg : x 2 Xg : Such a partition Q will be called a system of imprimitivity for (G; X). (ii) We say that the action (G; X) is primitive if it is not imprimitive. 1. (a) Suppose that (G; X; Q) is a system of imprimitivity for a transitive left action (G; X), Y 2 Q, y 2 Y . Let H = StabG(Y ), K = StabG(y). Prove the following statements. • K < H < G and H operates transitively on Y . • jXj = jY j · jQj, jQj = [G : H], jY j = [H : K]. (These statements hold even if X is infinite.) (b) Suppose that (G; X) is a transitive left action, y 2 X, K := StabG(y), and H is a subgroup of G such that K < H < G. Let Y := H · y ⊂ X, and let Q := fg · Y j g 2 Gg. (Our general notation scheme is that StabG(Y ) := fg 2 G j g · Y = Y g.) Show that (G; X; Q) is a system of imprimitivity. (c) Suppose that (G; X) is a transitive left action, x 2 X. Show that (G; X) is primitive if and only if Gx is a maximal proper subgroup of G. -
Computing the Maximal Subgroups of a Permutation Group I
Computing the maximal subgroups of a permutation group I Bettina Eick and Alexander Hulpke Abstract. We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a “hybrid group” approach; that is, we first compute a large solvable normal subgroup of the given permutation group and then use this to split the computation in various parts. 1991 Mathematics Subject Classification: primary 20B40, 20-04, 20E28; secondary 20B15, 68Q40 1. Introduction Apart from being interesting themselves, the maximal subgroups of a group have many applications in computational group theory: They provide a set of proper sub- groups which can be used for inductive calculations; for example, to determine the character table of a group. Moreover, iterative application can be used to investigate parts of the subgroups lattice without the excessive resource requirements of com- puting the full lattice. Furthermore, algorithms to compute the Galois group of a polynomial proceed by descending from the symmetric group via a chain of iterated maximal subgroups, see [Sta73, Hul99b]. In this paper, we present a new approach towards the computation of the conju- gacy classes of maximal subgroups of a finite permutation group. For this purpose we use a “hybrid group” method. This type of approach to computations in permu- tation groups has been used recently for other purposes such as conjugacy classes [CS97, Hul], normal subgroups [Hul98, CS] or the automorphism group [Hol00]. For finite solvable groups there exists an algorithm to compute the maximal sub- groups using a special pc presentation, see [CLG, Eic97, EW].