A characterization of the quaternion group (original) (raw)
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Some Properties of Representation of Quaternion Group
KnE Engineering
The quaternions are a number system in the form + + +. The quaternions ±1, ± , ± , ± form a non-abelian group of order eight called quaternion group. Quaternion group can be represented as a subgroup of the general linear group 2 (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.
Journal of Algebra, 2002
Two theorems are proved, the first of them showing that a modular quaternion-free finite 2-group has a characteristic abelian subgroup with metacyclic factor, the second classifying nonmodular finite quaternion-free 2-groups.
On infinite groups generated by two quaternions
Eprint Arxiv Math 0502512, 2005
Let x, y be two integral quaternions of norm p and l, respectively, where p, l are distinct odd prime numbers. We investigate the structure of x, y , the multiplicative group generated by x and y. Under a certain condition which excludes x, y from being free or abelian, we show for example that x, y , its center, commutator subgroup and abelianization are finitely presented infinite groups. We give many examples where our condition is satisfied and compute as an illustration a finite presentation of the group 1+j+k, 1+2j having these two generators and seven relations. In a second part, we study the basic question whether there exist commuting quaternions x and y for fixed p, l, using results on prime numbers of the form r 2 + ms 2 and a simple invariant for commutativity.
Complete decomposition of the generalized quaternion groups
Open Mathematics
Let G G be a finite nonabelian group. For any integer m ≥ 2 m\ge 2 , let A 1 , … , A m {A}_{1},\ldots ,{A}_{m} be nonempty subsets of G G . If A 1 , … , A m {A}_{1},\ldots ,{A}_{m} are mutually disjoint and if the subset product A 1 … A m = { α 1 … α m ∣ α v ∈ A v , v = 1 , 2 , … , m } {A}_{1}\ldots {A}_{m}=\left\{{\alpha }_{1}\ldots {\alpha }_{m}| {\alpha }_{v}\in {A}_{v},v=1,2,\ldots ,m\right\} coincides with G G , then ( A 1 , … , A m ) \left({A}_{1},\ldots ,{A}_{m}) is called a complete decomposition of G G of order m m . In this article, we let G G be the generalized quaternion groups Q 2 n {Q}_{{2}^{n}} , which is a finite nonabelian group of order 2 n {2}^{n} with group presentation given by ⟨ x , y ∣ x 2 n − 1 = 1 , y 2 = x 2 n − 2 , y x = x 2 n − 1 − 1 y ⟩ \langle x,y| {x}^{{2}^{n-1}}=1,{y}^{2}={x}^{{2}^{n-2}},yx={x}^{{2}^{n-1}-1}y\rangle for positive integer n ≥ 3 n\ge 3 . We determine the existence of complete decompositions of Q 2 n {Q}_{{2}^{n}} of order k k , for k ∈ {...
The Exhaustion Numbers of the Generalized Quaternion Groups
Malaysian Journal of Mathematical Sciences, 2023
Let G be a finite group and let T be a non-empty subset of G. For any positive integer k, let T k = {t1. .. t k | t1,. .. , t k ∈ T }. The set T is called exhaustive if T n = G for some positive integer n where the smallest positive integer n, if it exists, such that T n = G is called the exhaustion number of T and is denoted by e(T). If T k ̸ = G for any positive integer k, then T is a nonexhaustive subset and we write e(T) = ∞. In this paper, we investigate the exhaustion numbers of subsets of the generalized quaternion group Q2n = ⟨x, y | x 2 n−1 = 1, x 2 n−2 = y 2 , yx = x 2 n−1 −1 y⟩ where n ≥ 3. We show that Q2n has no exhaustive subsets of size 2 and that the smallest positive integer k such that any subset T ⊆ Q2n of size greater than or equal to k is exhaustive is 2 n−1 + 1. We also show that for any integer k ∈ {3,. .. , 2 n }, there exists an exhaustive subset T of Q2n such that |T | = k.
Quaternionic root systems and subgroups of the Aut"F 4
JOURNAL OF MATHEMATICAL PHYSICS, 2006
Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions, and octonions. Starting with the roots and weights of SU2 expressed as the real numbers one can construct the root systems of the Lie algebras of SO4 , SP2SO5 , SO8 , SO9 , F 4 and E 8 in terms of the discrete elements of the division algebras. The roots themselves display the groups structures besides the octonionic roots of E 8 which form a closed octonion algebra. The automorphism group AutF 4 of the Dynkin diagram of F 4 of order 2304, the largest crystallographic group in four-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F 4. The Weyl groups of many Lie algebras, such as, G 2 , SO7 , SO8 , SO9 , SU3XSU3, and SP3 SU2 have been constructed as the subgroups of AutF 4. We have also classified the other non-parabolic subgroups of AutF 4 which are not Weyl groups. Two subgroups of orders 192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group G 2 of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO8 manifesting itself as the cyclic symmetry of the quaternionic imaginary units e 1 , e 2 , e 3 is used to show that SO7 and SO9 can be embedded, triply symmetric way in SO8 and F 4 in respectively.
Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions
Geometriae Dedicata, 2017
Using the rings of Lipschitz and Hurwitz integers H(Z) and Hur(Z) in the quaternion division algebra H, we define several Kleinian discrete subgroups of P SL(2, H). We define first a Kleinian subgroup P SL(2, L) of P SL(2, H(Z)). This group is a generalization of the modular group P SL(2, Z). Next we define a discrete subgroup P SL(2, H) of P SL(2, H) which is obtained by using Hurwitz integers and in particular the subgroup of order 24 consisting of Hurwitz units. It contains as a subgroup P SL(2, L). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic half space H 1 H := {q ∈ H : (q) > 0}. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients H 1 H /P SL(2, L) and H 1 H /P SL(2, H) which are quaternionic versions of the classical modular orbifold and they are of finite volume. We give a thorough study of the Iwasawa decompositions, affine subgroups, and their descriptions by Lorentz transformations in the Lorentz-Minkowski model of hyperbolic 4-space. We give abstract finite presentations of these modular groups in terms of generators and relations via the Cayley graphs associated to the fundamental domains. We also describe a set of Selberg covers (corresponding to finite-index subgroups acting freely) which are quaternionic hyperbolic manifolds of finite volume with cusps whose sections are 3-tori. These hyperbolic arithmetic 4-manifolds are topologically the complement of linked 2-tori in the 4-sphere, in analogy with the complement in the 3-sphere of the Borromean rings and are related to the ubiquitous hyperbolic 24-cell. Finally we study the Poincaré extensions of these Kleinian groups to arithmetic Kleinian groups acting on hyperbolic 5-space and described in the quaternionic setting. In particular P SL(2, H(Z)) and P SL(2, Hur(Z)) are discrete subgroups of isometries of H 5 R and H 5 R /P SL(2, H(Z)), H 5 R /P SL(2, Hur(Z)) are examples of arithmetic 5-dimensional hyperbolic orbifolds of finite volume.
Finite groups with Quaternion Sylow subgroup
Comptes Rendus Mathématique, 2021
In this paper we show that a finite group G with Quaternion Sylow 2 -subgroup is 2 -nilpotent if, either 3 - |G| or G is solvable and the order of its Sylow 2-subgroup is strictly greater than 16