Quaternion Group Research Articles

Blocks with Defect Group $\boldsymbol{Q_{2^n}\times C_{2^m}}$ and $\boldsymbol{SD_{2^n}\times C_{2^m}}$

We determine the numerical invariants of blocks with defect group \(Q_{2^n}\times C_{2^m}\) and \(SD_{2^n}\times C_{2^m}\), where \(Q_{2^n}\) denotes a quaternion group of order 2n, \(C_{2^m}\) denotes a cyclic group of order 2m, and \(SD_{2^n}\) denotes a semidihedral group of order 2n. This generalizes Olsson’s results for m = 0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).

The optimal constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group

We determine the best (optimal) constant in the L2 Folland-Stein inequality on the quaternionic Heisenberg group and the non-negative functions for which equality holds

A characterization of subgroup containment in direct products

Using Goursat’s characterization of subgroups in a direct product, a characterization of subgroup containment in such a direct product is given. Applying this along with facts about p-groups we describe a process for obtaining the subgroup lattice of a direct product of p-groups. This process is implemented to portray the subgroup lattice of Q × Q, where Q is the quaternion group of order 8.

On the Elastic Energy Density of Constrained Q-Tensor Models for Biaxial Nematics

Within the Landau–de Gennes theory, the order parameter describing a biaxial nematic liquid crystal assigns a symmetric traceless 3 × 3 matrix Q with three distinct eigenvalues to every point of the region Ω occupied by the system. In the constrained case of matrices Q with constant eigenvalues, the order parameter space is diffeomorphic to the eightfold quotient \({\mathbb{S}^3/\mathcal{H}}\) of the 3-sphere \({\mathbb{S}^3}\), where \({\mathcal{H}}\) is the quaternion group, and a configuration of a biaxial nematic liquid crystal is described by a map from Ω to \({\mathbb{S}^3/\mathcal{H}}\). We express the (simplest form of the) Landau–de Gennes elastic free-energy density as a density defined on maps \({q: \Omega \to \mathbb{S}^3}\), whose functional dependence is restricted by the requirements that (1) it is well defined on the class of configuration maps from Ω to \({\mathbb{S}^3/\mathcal{H}}\) (residual symmetry) and (2) it is independent of arbitrary superposed rigid rotations (frame indifference). As an application of this representation, we then discuss some properties of the corresponding energy functional, including coercivity, lower semicontinuity and strong density of smooth maps. Other invariance properties are also considered. In the discussion, we take advantage of the identification of \({\mathbb{S}^3}\) with the Lie group of unit quaternions \({Sp(1) \cong SU(2)}\) and of the relations between quaternions and rotations in \({\mathbb{R}^3}\) and \({\mathbb{R}^4}\).

The Rational valued characters of the Group (Q2m×C2)

The main purpose of this paper is to find the rational valued characters table of the group (Q2m×C2),when m=2h , h             Z+, which is denoted by ≡*(Q2h+1×C2) ,where Q2m is denoted to Quaternion group ,and C2 is the cyclic group of order 2.Moreover we have found the general form of the rational valued characters table of the group (Q2h+1×C2).

A new linear quotient of C 4 admitting a symplectic resolution

We show that the quotient C 4/G admits a symplectic resolution for $${G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}$$ . Here Q 8 is the quaternionic group of order eight and D 8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements −Id of each. It is equipped with the tensor product representation $${{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}$$ . This group is also naturally a subgroup of the wreath product group $${Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}$$ . We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C 4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.

Cellular decomposition of quaternionic spherical space forms

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group.

HIGHER INDICATORS FOR SOME GROUPS AND THEIR DOUBLES

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for the Drinfel'd doubles of these groups. This first family of groups include the dihedral groups, the non-abelian groups of order pq, and the semidihedral groups. We find that the indicators are all integers, with negative integers being possible in the first family only under certain specific conditions.

EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

Finite groups with S-supplemented p-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by BsG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ BsG. A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.

On the Structure of Augmentation Quotient Groups for the Generalized Quaternion Group

For the generalized quaternion group G, this article deals with the problem of presenting the nth power Δn(G) of the augmentation ideal Δ (G) of the integral group ring ZG. The structure of Qn(G)=Δn(G)/Δn+1(G) is obtained.

The modular S-matrix as order parameter for topological phase transitions

We study topological phase transitions in discrete gauge theories in two spatial dimensions induced by the formation of a Bose condensate. We analyse a general class of euclidean lattice actions for these theories which contain one coupling constant for each conjugacy class of the gauge group. To probe the phase structure we use a complete set of open and closed anyonic string operators. The open strings allow one to determine the particle content of the condensate, whereas the closed strings enable us to determine the matrix elements of the modular S-matrix, in both the unbroken and broken phases. From the measured broken S-matrix we may read off the sectors that split or get identified in the broken phase, as well as the sectors that are confined. In this sense the modular S-matrix can be employed as a matrix valued nonlocal order parameter from which the low-energy effective theories that occur in different regions of parameter space can be fully determined. To verify our predictions, we studied a non-abelian anyon model based on the quaternion group of the order of eight by Monte Carlo simulation. We probe part of the phase diagram for the pure gauge theory and find a variety of phases with magnetic condensates leading to various forms of (partial) confinement in complete agreement with the algebraic breaking analysis. Also the order of various transitions is established.

On square roots of the Haar state on compact quantum groups

The paper is concerned with the extension of the classical study of probability measures on a compact group which are square roots of the Haar measure, due to Diaconis and Shahshahani, to the context of compact quantum groups. We provide a simple characterisation for compact quantum groups which admit no non-trivial square roots of the Haar state in terms of their corepresentation theory. In particular it is shown that such compact quantum groups are necessarily of Kac type and their subalgebras generated by the coefficients of a fixed two-dimensional irreducible corepresentation are isomorphic (as finite quantum groups) to the algebra of functions on the group of unit quaternions. An example of a quantum group whose Haar state admits no nontrivial square root and which is neither commutative nor cocommutative is given.

A Bound on the Order of Non-Abelian Tensor Square of a Prime-Power Group

This article improves on an upper bound for the order of the non-abelian tensor square of a finite p-group G given in [3]. In particular, applying this for finite p-groups of order p n with factor group G/G′ of order p m , the bound p nm attains if and only if G is elementary abelian of rank n, quaternion group of order 8, or extra special p-group of order p 3 with odd exponent p.

On seven-dimensional quaternionic contact solvable Lie groups

Abstract We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven-dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven-dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven-dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.

Topological Quantum Liquids with Quaternion Non-Abelian Statistics

Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).

On the Periods of 2‐Step General Fibonacci Sequences in the Generalized Quaternion Groups

We study 2‐step general Fibonacci sequences in the generalized quaternion groups Q4n. In cases where the sequences are proved to be simply periodic, we obtain the periods of 2‐step general Fibonacci sequences.

FACTORIZATION NUMBERS OF SOME FINITE GROUPS

AbstractFor a finite group G, let F2(G) be the number of factorizations G = AB of the group G, where A and B are subgroups of G. We compute F2(G) for certain classes of groups, including cyclic groups ℤn, elementary abelian p-groups ℤpn, dihedral groups D2n, generalised quaternion groups Q4n, quasi-dihedral 2-groups QD2n(n≥4), modular p-groups Mpn, projective general linear groups PGL(2, pn) and projective special linear groups PSL(2, pn).

Conformal flatness, non-Abelian Kaluza–Klein reduction and quaternions

The non-Abelian Kaluza–Klein reduction of conformally flat spaces is considered for arbitrary dimensions and signatures. The corresponding equations are particularly elegant when the internal space supports a global Killing parallelization. Assuming this imposes the generalized ‘spacetime’ to be maximally symmetric with holonomy in the unitary quaternionic group S p ( d / 4 ) . Recalling an analogous result for the complex case, we conclude that all special manifolds with constant properly ‘holonomy-related’ sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza–Klein spaces.

Degenerate principal series representations for quaternionic unitary groups

We give a complete description of all points of reducibility and the composition series of the degenerate principal series representations for quaternionic unitary groups which are induced from a character of the maximal parabolic subgroup with abelian unipotent radical. The case of even orthogonal groups is also included.