A Classification of Clifford Algebras As Images of Group Algebras of Salingaros Vee Groups
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The Homology of Peiffer Products of Groups
New York Journal of Mathematics New York J. Math. 6 (2000) 55–71. The Homology of Peiffer Products of Groups W. A. Bogley and N. D. Gilbert Abstract. The Peiffer product of groups first arose in work of J.H.C. White- head on the structure of relative homotopy groups, and is closely related to problems of asphericity for two-complexes. We develop algebraic methods for computing the second integral homology of a Peiffer product. We show that a Peiffer product of superperfect groups is superperfect, and determine when a Peiffer product of cyclic groups has trivial second homology. We also introduce a double wreath product as a Peiffer product. Contents Introduction 55 1. The low-dimensional homology of products of subgroups 57 2. Twisted bilinear relations 60 3. The structure of SG∗H 61 4. Computations 63 References 70 Introduction Given two groups acting on each other by automorphisms, it is natural to ask whether these groups can be embedded in an overgroup in such a way that the given actions are realized by conjugation. If the actions are trivial, this can be done simply by forming the direct product of the two groups. In general, the question has a negative answer. One is led to the following construction. Let G and H be groups and suppose we are given fixed actions (g, h) 7→ gh and (h, g) 7→ hg of each group on the other. Received October 1, 1999. Mathematics Subject Classification. 20J05, 20E22, 20F05. Key words and phrases. homology, Peiffer product, asphericity, two-complex, double wreath product. -
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]. -
Minimal Generation of Transitive Permutation Groups
Minimal generation of transitive permutation groups Gareth M. Tracey∗ Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom October 30, 2017 Abstract This paper discusses upper bounds on the minimal number of elements d(G) required to generate a transitive permutation group G, in terms of its degree n, and its order G . In particular, we | | reduce a conjecture of L. Pyber on the number of subgroups of the symmetric group Sym(n). We also prove that our bounds are best possible. 1 Introduction A well-developed branch of finite group theory studies properties of certain classes of permutation groups as functions of their degree. The purpose of this paper is to study the minimal generation of transitive permutation groups. For a group G, let d(G) denote the minimal number of elements required to generate G. In [21], [7], [26] and [28], it is shown that d(G)= O(n/√log n) whenever G is a transitive permutation group of degree n 2 (here, and throughout this paper, “ log ” means log to the base 2). A beautifully ≥ constructed family of examples due to L. Kov´acs and M. Newman shows that this bound is ‘asymp- totically best possible’ (see Example 6.10), thereby ending the hope that a bound of d(G)= O(log n) could be proved. The constants involved in these theorems, however, were never estimated. We prove: arXiv:1504.07506v3 [math.GR] 30 Jan 2018 Theorem 1.1. Let G be a transitive permutation group of degree n 2. Then ≥ (1) d(G) cn ,where c := 1512660 log (21915)/(21915) = 0.920581 . -
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. -
Bounds of Some Invariants of Finite Permutation Groups Hülya Duyan
Bounds of Some Invariants of Finite Permutation Groups H¨ulya Duyan Department of Mathematics and its Applications Central European University Budapest, Hungary CEU eTD Collection A dissertation presented for the degree of Doctor of Philosophy in Mathematics CEU eTD Collection Abstract Let Ω be a non-empty set. A bijection of Ω onto itself is called a permutation of Ω and the set of all permutations forms a group under composition of mapping. This group is called the symmetric group on Ω and denoted by Sym(Ω) (or Sym(n) or Sn where jΩj = n). A permutation group on Ω is a subgroup of Sym(Ω). Until 1850's this was the definition of group. Although this definition and the ax- iomatic definition are the same, usually what we first learn is the axiomatic approach. The reason is to not to restrict the group elements to being permutations of some set Ω. Let G be a permutation group. Let ∼ be a relation on Ω such that α ∼ β if and only if there is a transformation g 2 G which maps α to β where α; β 2 Ω. ∼ is an equivalence relation on Ω and the equivalence classes of ∼ are the orbits of G. If there is one orbit then G is called transitive. Assume that G is intransitive and Ω1;:::; Ωt are the orbits of G on Ω. G induces a transitive permutation group on each Ωi, say Gi where i 2 f1; : : : ; tg. Gi are called the transitive constituents of G and G is a subcartesian product of its transitive constituents. -
Math.AG] 17 Mar 2001 Ebro Uhafml Mle H Og Ojcuefrinfin field 5-Dimensional for Such with Identify Type Conjecture to Weil Hope Hodge We of the Algebra
QUATERNIONIC PRYMS AND HODGE CLASSES B. VAN GEEMEN AND A. VERRA Abstract. Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n 1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomolo−gy group. In general, it is not clear that these are cycle classes. In this paper we show that a particular 6-dimensional family of such 8-folds are Prym varieties and we use the method of C. Schoen to show that all Hodge classes on the general abelian variety in this family are algebraic. We also consider Hodge classes on certain 5-dimensional subfamilies and relate these to the Hodge conjecture for abelian 4-folds. In this paper we study abelian varieties of dimension 8 whose endomorphism ring is a definite quaternion algebra over Q, we refer to these as abelian 8-folds of quaternion type. Such abelian varieties are interesting since their Hodge rings are not generated by divisor classes [M] and the Hodge conjecture is still open for most of them. The moduli space of 8-folds of quaternion type, with fixed discrete data, is 6-dimensional. One such moduli space was investigated recently by Freitag and Hermann [FrHe]. In [KSTT] a 6-dimensional family of K3 surfaces is studied whose Kuga-Satake varieties have simple factors of dimension 8 which are of quaternion type. Our first result, in section 3, is that one of these moduli spaces parametrizes Prym varieties of unramified 2:1 covers C˜ Cˆ, actually the genus 17 curves C˜ are 8:1 unramified covers of genus 3 curves with Galois→ group Q generated by the quaternions i and j. -
A Note on the Quaternion Group As Galois Group
proceedings of the american mathematical society Volume 108, Number 3, March 1990 A NOTE ON THE QUATERNION GROUP AS GALOIS GROUP ROGER WARE (Communicated by Louis J. Ratliff, Jr.) Abstract. The occurrence of the quaternion group as a Galois group over cer- tain fields is investigated. A theorem of Witt on quaternionic Galois extensions plays a key role. In [9, §6] Witt proved a theorem characterizing quaternionic Galois exten- sions. Namely, he showed that if F is a field of characteristic not 2 then an extension L = F(y/ä,y/b), a,b e F, of degree 4 over F can be embedded in a Galois extension A" of 7 with Gal(K/F) = 77g (the quaternion group 7 7 7 of order 8) if and only if the quadratic form ax + by + abz is isomorphic to x 2 + y 2 + z 2 .In addition he showed how to explicitly construct the Galois extension from the isometry. An immediate and interesting consequence of this is the fact that 77g cannot be a Galois group over any Pythagorean field. In this note Witt's theorem is used to obtain additional results about the existence of 77g as a Galois group over certain fields. If 7 is a field (of char- acteristic not 2) with at most one (total) ordering such that 77g does not oc- cur as a Galois group over F then the structure of the pro-2-Galois groups GF(2) = Gal(7(2)/7), Gpy = Gal(7py/7) (where 7(2) and Fpy are the quadratic and Pythagorean closures of F) are completely determined. -
A Survey on Automorphism Groups of Finite P-Groups
A Survey on Automorphism Groups of Finite p-Groups Geir T. Helleloid Department of Mathematics, Bldg. 380 Stanford University Stanford, CA 94305-2125 [email protected] February 2, 2008 Abstract This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presented with varying levels of detail, with some of the examples given in complete detail. 1 Introduction The goal of this survey is to communicate some of what is known about the automorphism groups of finite p-groups. The focus is on three topics: explicit computations for familiar finite p-groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. Section 2 begins with some general theorems on automorphisms of finite p-groups. Section 3 continues with explicit examples of automorphism groups of finite p-groups found in the literature. This arXiv:math/0610294v2 [math.GR] 25 Oct 2006 includes the computations on the automorphism groups of the extraspecial p- groups (by Winter [65]), the Sylow p-subgroups of the Chevalley groups (by Gibbs [22] and others), the Sylow p-subgroups of the symmetric group (by Bon- darchuk [8] and Lentoudis [40]), and some p-groups of maximal class and related p-groups. -
On Sylow 2-Subgroups of Finite Simple Groups of Order up to 210
ON SYLOW 2-SUBGROUPS OF FINITE SIMPLE GROUPS OF ORDER UP TO 210 DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Sergey Malyushitsky, M.S. ***** The Ohio State University 2004 Dissertation Committee: Approved by Professor Koichiro Harada, Adviser Professor Akos Seress Adviser Professor Ronald Solomon Department of Mathematics ABSTRACT A 2-group ( a group of order a power of 2 ) is called realizable if it occurs as a Sylow 2-subgroup of a finite simple group. The purpose of this thesis is to study all realizable groups of order up to 210. From the classification of all simple groups of finite order we know all realizable groups of order up to 210 as Sylow 2-subgroups of known finite simple groups. However without the use of classification determining all realizable 2-groups is very difficult. In the first part of the thesis we present an argument that produces all realizable groups of order up to 32, by eliminating one by one all 2-groups that can not occur as a Sylow 2-subgroup of a finite simple group. When the number of 2-groups of given order becomes too large to handle it by hand we attempt to use a computer for repetitive checks on a large number of 2-groups. The second part of the thesis is devoted to describing all realizable 2-groups of order up to 210 using the classification of all finite simple groups. We determine the identification number of each group in the Small Groups Library in GAP4 and compute the power-commutator presentation of each realizable group S of type G. -
Some Properties of Representation of Quaternion Group
ICBSA 2018 International Conference on Basic Sciences and Its Applications Volume 2019 Conference Paper Some Properties of Representation of Quaternion Group Sri Rahayu, Maria Soviana, Yanita, and Admi Nazra Department of Mathematics, Andalas University, Limau Manis, Padang, 25163, Indonesia Abstract The quaternions are a number system in the form 푎 + 푏푖 + 푐푗 + 푑푘. The quaternions ±1, ±푖, ±푗, ±푘 form a non-abelian group of order eight called quaternion group. Quaternion group can be represented as a subgroup of the general linear group 퐺퐿2(C). In this paper, we discuss some group properties of representation of quaternion group related to Hamiltonian group, solvable group, nilpotent group, and metacyclic group. Keywords: representation of quaternion group, hamiltonian group, solvable group, nilpotent group, metacyclic group Corresponding Author: Sri Rahayu [email protected] Received: 19 February 2019 1. Introduction Accepted: 5 March 2019 Published: 16 April 2019 First we review that quaternion group, denoted by 푄8, was obtained based on the Publishing services provided by calculation of quaternions 푎 + 푏푖 + 푐푗 + 푑푘. Quaternions were first described by William Knowledge E Rowan Hamilton on October 1843 [1]. The quaternion group is a non-abelian group of Sri Rahayu et al. This article is order eight. distributed under the terms of the Creative Commons Attribution License, which 2. Materials and Methods permits unrestricted use and redistribution provided that the original author and source are Here we provide Definitions of quaternion group dan matrix representation. credited. Selection and Peer-review under Definition 1.1. [2] The quaternion group, 푄8, is defined by the responsibility of the ICBSA Conference Committee. -
Random Generation of Finite and Profinite Groups and Group
Annals of Mathematics 173 (2011), 769{814 doi: 10.4007/annals.2011.173.2.4 Random generation of finite and profinite groups and group enumeration By Andrei Jaikin-Zapirain and Laszl´ o´ Pyber Abstract We obtain a surprisingly explicit formula for the number of random ele- ments needed to generate a finite d-generator group with high probability. As a corollary we prove that if G is a d-generated linear group of dimension n then cd + log n random generators suffice. Changing perspective we investigate profinite groups F which can be generated by a bounded number of elements with positive probability. In response to a question of Shalev we characterize such groups in terms of certain finite quotients with a transparent structure. As a consequence we settle several problems of Lucchini, Lubotzky, Mann and Segal. As a byproduct of our techniques we obtain that the number of r-relator groups of order n is at most ncr as conjectured by Mann. 1. Introduction Confirming an 1882 conjecture of Netto [40], Dixon [13] proved in 1969 that two randomly chosen elements generate the alternating group Alt(n) with probability that tends to 1 as n ! 1. This was extended in [21] and [24] to arbitrary sequences of non-abelian finite simple groups. Such results form the basis of applying probabilistic methods to the solution of various problems concerning finite simple groups [50]. Interest in random generation of more general families of finite groups arose when it was realized that randomized algorithms play a critical role in handling matrix groups [4]. -
Math 371 Review 1
Math 371 Review 1 Some Notation and Basic Concepts Linear Algebra fields. • Examples of fields: • { the field of rational, real and complex numbers: Q, R and C { Fp := the finite field with p elements, where p is a prime number { the field of rational functions, Q(x), R(x), C(x), consisting of quotients of poly- nomials with coefficients in Q, R, C respectively. vector spaces, subspaces. • dimension, basis, linear independence. • linear transformations (also called \homomorphisms between vector spaces") • matrix representations of a linear transformation. (So, after choosing a basis of the • source and a basis of a target vector space, a linear transformation is represented by a matrix.) Effect of change of bases on the matrix representation. direct sum of vector spaces • eigenvalues, eigenvectors, and eigenspaces (of a linear transformation), characteristic • polynomials. diagonalization (of a linear transformation) • Basic Grooup Theory groups, subgroups • Examples of groups • { cyclic groups: Z :=integers, Z=nZ = a cyclic group with n elements. { the dihedral group D2n with 2n elements; it is the group of all symmetries of a regular n-gon. { the quaternion group Q8 with 8 elements. { the \general linear groups" GLn(Q), GLn(R), GLn(C), GLn(Fp). These are the group of invertible n n matrices with coefficients in Q, R, C, and Fp respec- × tively. Notice that GL1(Q) = Q×, the group of non-zero rational numbers under multiplication; similarly GL1(R) = R×, GL1(C) = C×, GL1(Fp) = Fp×. 1 { the \special linear groups" SLn(Q), SLn(R), SLn(C) and SLn(Fp); each is the subgroup of the respective general linear group consisting of all elements whose determinant is equal to 1.