DOCSLIB.ORG

The Arithmetic of Quaternion Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Arithmetic of Quaternion Algebra The Arithmetic of Quaternion Algebra M-F Vigneras August 2, 2006 2 Contents 1 Quaternion Algebra over a ¯eld 1 1.1 Quaternion Algebra . 1 1.2 The theorem of automorphisms and the neutralizing ¯elds . 5 1.3 Geometry . 8 1.4 Orders and ideals . 13 2 Quaternion algebra over a local ¯eld 23 2.1 Classi¯cation . 23 2.2 Study of M(2;K)........................... 28 2.3 Orders embedded maximally . 32 2.4 Zeta function . 35 3 Quaternion algebra over a global ¯eld 43 3.1 Adeles . 43 3.2 Zeta function, Tamagawa number . 48 3.3 Cassi¯cation . 57 3.4 Norm theorem and strong approximation theorem . 61 3.5 Orders and ideals . 63 4 Applications to Arithmetic Groups 79 4.1 Quaternion groups . 79 4.2 Riemann surfaces . 85 4.3 Examples and Applications . 92 5 Quaternion arithmetic in the case where the Eichler condition is not satis¯ed any more 105 5.1 Units . 106 5.2 Class number . 108 5.3 Examples . 111 i ii CONTENTS Chapter 1 Quaternion Algebra over a ¯eld In this chapter K always denotes a commutative ¯eld of arbitrary characteristics if no particular mention, and Ks is the separable closure of K. 1.1 Quaternion Algebra De¯nition 1.1. A quaternion algebra H with center K is a central algebra over K of dimension 4, such that there exists a separable algebra of dimension 2 over K, and an invertible element θ in K with H = L + Lu, where u 2 H satis¯es u2 = θ; um = mu (1.1) for all m 2 L, and where m 7! m is the nontrivial K-automorphism of L. We denote H sometimes by fL; θg, but H does not determine this pair fL; θg uniquely. For example, it is clear that one could replace θ by θmm, if m is an element of L such that mm 6= 0. the element u is not determined by (1). If m 2 L is an element with mm = 1, we could replace u by mu. the de¯nition can be used in the case of arbitrary character. .we can verify easily that HK is a central simple algebra, i.e. an algebra with center K and without any nontrivial two-sided ideal.conversely, we can prove that every cental simple algebra of dimension 4 over K is a quaternion algebra.The rule of multiplication is deduced from (1). If m 2 L, we have (m1 + m2u)(m3 + m4u) = (m1m3 + m2m4θ) + (m1m4 + m2m3)u: (1.2) De¯nition 1.2. The conjugation is the K-endomorphism of H: h ! h, which is the extension of the nontrivial K-automorphism of L de¯ned by u = ¡u. It is easy to verify it is an involutive anti-automorphism of H. It can be expressed by the following relations: if h; k 2 H and a; b 2 K, we have ah + bk = ah + bk; h = h; hk = kh: De¯nition 1.3. Assume h 2 H. The reduced trace of h is t(h) = h + h. The reduced norm of h is n(h) = hh. 1 2 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD If h2 = K, its' minimal polynomial on K is (x ¡ h)(x ¡ h) = x2 ¡ t(h)x + n(h) The algebra K(h), generated by h over K, is a quadratic extension over K. The reduced trace and the reduced norm of h are simply the images of h by the operations trace and norm on K(h)=K. The conjugation and the identity are the K-automorphism of K(h). Under the usual de¯nition of the trace and the norm of a K-algebra [cf.Bourbaki[1]], the trace of H=K is T = 2t, The norm of H=K is N = 2n. We denote the group of the units in a ring X by X£. Lemma 1.1.1. The invertible elements in H are that with their reduced norm nontrivial. The reduced norm de¯nes a multiplicative homomorphism from H£ into K£. The reduced trace is K-linear, and the mapping (h; k) 7! t(hk) is a bilinear form non-degenerated on H. Proof. We leave the veri¯cation of the following very simple properties to reader as an exercise: n(hk) = n(h)n(k) n(h) 6= 0 is equivalent to The invertibility of h , and in this case h¡1 = hn(h)¡1, t(ah + bk) = at(h) + bt(k); t(hk) = t(kh), if a; b 2 K; andh; k 2 H. The fact that the mapping (h; k) ! t(hk) being non-degenerate comes from the assumption that L=K has been separable. In fact, if t(hk) = 0 whatever k 2 H , we have t(m1m = 0 for every m 2 L, if h = m1 + m2u,then m1 = 0. Similarly t(m2m = 0 for every m 2 L, whence m2 = 0 and h = 0. we note that, one of the advantage of the reduced trace is that, in the case of characteristic 2 the trace T = 2t is zero, but the reduced trace is non-degenerate. In case characteristic not 2, we recover the classical de¯nition of the quaternion algebra. the couple fL; θg is equivalent with a couple fa; bg formed by two nontrivial elements a; b in K and the relations (1) are determine H as the the K-algebra with basis 1; i; j; ij, where the elements i; j 2 H satisfy i2 = a; j2 = b; ij = ¡ji: (1.3) The transition between(1)and(3) carries out for example by setting L = K(i); θ = b; u = j. Setting k = ij one can write the table of multiplication of i; j; k , which shows these three elements acting symmetrically . The entities in the table are the products hh0: hn h' t j k i a k -j j -k b i k j -i -ab The conjugation,the reduced trace, and the reduced norm have their expression as follows: if h = x + yi + zj + tk, then h = x ¡ yi ¡ zj ¡ tk; t(h) = 2x; and n(h) = x2 ¡ ay2 ¡ bz2 + abt2; the coe±cient of k in h should not be confused with the reduced trace. We notice the another important property: the reduced norm de¯nes a quadratic form on the K-vector space V of the subjacent H. 1.1. QUATERNION ALGEBRA 3 We shall see that, the quaternion algebra H is de¯ned by the relations(1) or (3) to the forms fL; θg or fa; bg when the case is permitted. we also shall consider these notation u; i; j; t(h); h as the standard notation. The fundamental example of a quaternion algebra over K is given by the algebra M(2; k), the matrices of 2 with entities in K. The reduced trace and the reduced norm are the trace and the determinant as the usual sense in M(2;K). It can be identify K with its image in M(2;K) of the K-homomorphismµ ¶ which convert a b the unit of K to the identity matrix. Explicitly if h = 2 M(2;K), µ ¶ c d d ¡b h = , t(h) = a + d, n(h) = ad ¡ bc . We are going to show that ¡c a M(2; k) satis¯es the de¯nition of a quaternion algebra as follows. We choose a matrix with a distinguish value, and set L = K(m). Since m has the same distinguish value as m , it is similar to m.: there exists then an u 2 GL(2; k) such that umu¡1 = m. We verify that t(u) = 0, since t(um) = t(u)m 2 K for each m 2 L,from this we deduce u2 = θ 2 K0. In the following remark we are likely going to explain why is M(2;K) the fundamental example: Over a separably closed ¯eld, M(2;K) is the unique quaternion algebra up to an isomorphism. In fact, every separable algebra of dimension 2 over K can not be a ¯eld sent by the norm on K£ surjectively, and being included in M(2; k)(an inclusion is an injective K-homomorphism). From this we derive that, it is isomorphic to fK + K; 1g ' M(2;K) thanks to the realization of M(2;K) as a quaternion algebra done above. tensor product. Let F be a commutative ¯eld containing K. We verify directly by the de¯nition the tensor product of a quaternion algebra with F over K is a quaternion algebra over F , and that F ­ fL; θg = fF ­ L; θg . We write the obtained quaternion algebra by HF too. The algebra H is included in HF in a natural way. Taking the separable closure Ks of K as F we see that H is included in M(2;Ks). De¯nition 1.4. The ¯elds F=K such that HF to be isomorphic to M(2;F ) is called the neutralized ¯elds of H in M(2; k). The inclusions of H in M(2;F ) is called the F -representations. Examples. (1) The quaternion algebra over K has no any K-representation if it is not isomorphic to M(2;K). (2)We de¯neµ the¶ followingµ matrices:¶ µ ¶ 1 0 0 1 0 1 I = , J = , IJ = : These matrices satisfy the 0 ¡1 1 0 ¡1 0 relations (3) with a = b = 1. We derived from them in the case of characteristic unequal to 2, a quaternion algebra fa; bg is isomorphic to µ p p p ¶ x + ay b(z + at f p p p jx; y; z; t 2 Kg; b(z ¡ at) x ¡ ay p p where a and b are two roots of a and b in Ks .
Load more

Recommended publications
  • Quaternions and Cli Ord Geometric Algebras
    Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a .
    [Show full text]
  • 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].
    [Show full text]
  • Arxiv:1001.0240V1 [Math.RA]
    Fundamental representations and algebraic properties of biquaternions or complexified quaternions Stephen J. Sangwine∗ School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom. Email: [email protected] Todd A. Ell† 5620 Oak View Court, Savage, MN 55378-4695, USA. Email: [email protected] Nicolas Le Bihan GIPSA-Lab D´epartement Images et Signal 961 Rue de la Houille Blanche, Domaine Universitaire BP 46, 38402 Saint Martin d’H`eres cedex, France. Email: [email protected] October 22, 2018 Abstract The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. 1 Introduction It is typical of quaternion formulae that, though they be difficult to find, once found they are immediately verifiable. J. L. Synge (1972) [43, p34] arXiv:1001.0240v1 [math.RA] 1 Jan 2010 The quaternions are relatively well-known but the quaternions with complex components (complexified quaternions, or biquaternions1) are less so. This paper aims to set out the fundamental definitions of biquaternions and some elementary results, which, although elementary, are often not trivial. The emphasis in this paper is on the biquaternions as an applied algebra – that is, a tool for the manipulation ∗This paper was started in 2005 at the Laboratoire des Images et des Signaux (now part of the GIPSA-Lab), Grenoble, France with financial support from the Royal Academy of Engineering of the United Kingdom and the Centre National de la Recherche Scientifique (CNRS).
    [Show full text]
  • 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.
    [Show full text]
  • Arithmetic of Hyperbolic 3-Manifolds
    Arithmetic of Hyperbolic 3-Manifolds Alan Reid Rice University Hyperbolic 3-Manifolds and Discrete Groups Hyperbolic 3-space can be defined as 3 H = f(z; t) 2 C × R : t > 0g dsE and equipped with the metric ds = t . Geodesics are vertical lines perpendicular to C or semi-circles perpendicular to C. Codimension 1 geodesic submanifolds are Euclidean planes in 3 H orthogonal to C or hemispheres centered on C. 3 The full group of orientation-preserving isometries of H can be identified with PSL(2; C). Abuse of notation We will often view Kleinian groups as groups of matrices! A discrete subgroup of PSL(2; C) is called a Kleinian group. 3 Such a group acts properly discontinuously on H . If Γ is torsion-free (does not contain elements of finite order) then Γ acts freely. 3 In this latter case we get a quotient manifold H =Γ and a 3 3 covering map H ! H =Γ which is a local isometry. 3 We will be interested in the case when H =Γ has finite volume. Note: If Γ is a finitely generated subgroup of PSL(2; C) there is always a finite index subgroup that is torsion-free. Examples 1. Let d be a square-freep positive integer, and Od the ring of algebraic integers in Q( −d). Then PSL(2; Od) is a Kleinian group. 3 Indeed H =PSL(2; Od) has finite volume but is non-compact. These are known as the Bianchi groups. d = 1 (picture by Jos Leys) 2. Reflection groups. Tessellation by all right dodecahedra (from the Not Knot video) 3.Knot Complements The Figure-Eight Knot Thurston's hyperbolization theorem shows that many knots have complements that are hyperbolic 3-manifolds of finite volume.
    [Show full text]
  • The Geometry and Topology of Arithmetic Hyperbolic 3-Manifolds
    The geometry and topology of arithmetic hyperbolic 3-manifolds Alan W. Reid∗ May 12, 2007 1 Introduction This paper is based on three lectures given by the author at the RIMS Symposium, “Topology, Com- plex Analysis and Arithmetic of Hyperbolic Spaces” held at the Research Institute for Mathematical Sciences, Kyoto University, in December 2006. The goal of the lectures was to describe recent work on understanding topological and geometric properties of arithmetic hyperbolic 3-manifolds, and the connections with number theory. This is the theme of the paper. Our discussion of topological aspects of arithmetic hyperbolic 3-orbifolds is motivated by the following central conjectures from 3-manifold topology: Conjecture 1.1. Let M be a closed hyperbolic 3-manifold, then M is virtually Haken; ie M has a finite sheeted cover N which contains embedded incompressible surface (necessarily of genus at least 2). Conjecture 1.2. Let M be a closed hyperbolic 3-manifold, then M has a finite sheeted cover N for which b1(N) > 0 (where b1(N) is the first Betti number of N). We can also rephrase Conjecture 1.2 to say that vb1(M) > 0 where vb1(M) is defined by vb1(M) = sup{b1(N) : N is a finite cover of M}, and is called the virtual first Betti number of M. Conjecture 1.3. Let M be a closed hyperbolic 3-manifold, then vb1(M) = ∞. Conjecture 1.4. Let M be a closed hyperbolic 3-manifold, then π1(M) is large; that is to say, some finite index subgroup of π1(M) admits a surjective homomorphism onto a non-abelian free group.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • A Classification of Clifford Algebras As Images of Group Algebras of Salingaros Vee Groups
    DEPARTMENT OF MATHEMATICS TECHNICAL REPORT A CLASSIFICATION OF CLIFFORD ALGEBRAS AS IMAGES OF GROUP ALGEBRAS OF SALINGAROS VEE GROUPS R. Ablamowicz,M.Varahagiri,A.M.Walley November 2017 No. 2017-3 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville, TN 38505 A Classification of Clifford Algebras as Images of Group Algebras of Salingaros Vee Groups Rafa lAb lamowicz, Manisha Varahagiri and Anne Marie Walley Abstract. The main objective of this work is to prove that every Clifford algebra C`p;q is R-isomorphic to a quotient of a group algebra R[Gp;q] modulo an ideal J = (1 + τ) where τ is a central element of order 2. p+q+1 Here, Gp;q is a 2-group of order 2 belonging to one of Salingaros isomorphism classes N2k−1;N2k; Ω2k−1; Ω2k or Sk. Thus, Clifford al- gebras C`p;q can be classified by Salingaros classes. Since the group algebras R[Gp;q] are Z2-graded and the ideal J is homogeneous, the quotient algebras R[G]=J are Z2-graded. In some instances, the isomor- ∼ phism R[G]=J = C`p;q is also Z2-graded. By Salingaros Theorem, the groups Gp;q in the classes N2k−1 and N2k are iterative central products of the dihedral group D8 and the quaternion group Q8, and so they are extra-special. The groups in the classes Ω2k−1 and Ω2k are central products of N2k−1 and N2k with C2 × C2, respectively. The groups in the class Sk are central products of N2k or N2k with C4. Two algorithms to factor any Gp;q into an internal central product, depending on the class, are given.
    [Show full text]
  • On the Tensor Product of Two Composition Algebras
    On The Tensor Product of Two Composition Algebras Patrick J. Morandi S. Pumplun¨ 1 Introduction Let C1 F C2 be the tensor product of two composition algebras over a field F with char(F ) =¡ 2. R. Brauer [7] and A. A. Albert [1], [2], [3] seemed to be the first math- ematicians who investigated the tensor product of two quaternion algebras. Later their results were generalized to this more general situation by B. N. Allison [4], [5], [6] and to biquaternion algebras over rings by Knus [12]. In the second section we give some new results on the Albert form of these algebras. We also investigate the F -quadric defined by this Albert form, generalizing a result of Knus ([13]). £ £ Since Allison regarded the involution ¢ = 1 2 as an essential part of the algebra C = C1 C2, he only studied automorphisms of C which are compatible with ¢ . In the last section we show that any automorphism of C that preserves a certain biquaternion subalgebra also is compatible with ¢ . As a consequence, if C is the tensor product of two octonion algebras, we show that C does not satisfy the Skolem-Noether Theorem. Let F be a field and C a unital, nonassociative F -algebra. Then C is a composition algebra if there exists a nondegenerate quadratic form n : C ¤ F such that n(x · y) = n(x)n(y) for all x, y ¥ C. The form n is uniquely determined by these conditions and is called the norm of C. We will write n = nC . Composition algebras only exist in ranks 1, 2, 4 or 8 (see [10]).
    [Show full text]
  • 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.
    [Show full text]
  • The Arithmetic Hyperbolic 3-Manifold of Smallest Volume
    THE ARITHMETIC HYPERBOLIC 3-MANIFOLD OF SMALLEST VOLUME Ted Chinburg * Eduardo Friedman ** Kerry N. Jones y Alan W. Reid yy Abstract. We show that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold. The next smallest one is the Meyerhoff manifold. 0. Introduction x A hyperbolic 3-manifold is a 3-manifold admitting a complete Riemannian metric all of whose sectional curvatures are 1. The universal cover of such a manifold can therefore be identified with the hyperbolic− 3-space, that is, the unique connected and simply connected hyperbolic 3-manifold. We denote by H3 = (z; t) C R t > 0 ; f 2 ⊕ j g the upper half space model of hyperbolic 3-space. With this convention, the full group of orientation-preserving isometries of H3 is simply PGL(2; C). A Kleinian group Γ is a discrete subgroup of PGL(2; C). Hence an orientable hyperbolic 3- manifold is the quotient of H3 by a torsion-free Kleinian group, since this acts properly discontinuously and freely on H3. If we relax the condition that Γ act freely, allowing Γ to contain torsion elements, we obtain a complete orientable hyperbolic 3-orbifold (cf. [Th] for further details). We will only be concerned with the case that M = H3=Γ has a hyperbolic structure of finite volume (we say Γ has finite covolume). Therefore in what follows, by a hyperbolic 3-manifold or 3-orbifold we shall always mean a complete orientable hyperbolic 3-manifold or 3-orbifold of finite volume. By the Mostow and Prasad Rigidity Theorems [Mo] [Pr], hyperbolic volume is a topological invariant of a hyperbolic 3-orbifold and is therefore a natural object to study.
    [Show full text]