Quaternion Group Research Articles

Erdős–Gyárfás conjecture for some families of Cayley graphs

The Paul Erdős and Andras Gyarfas conjecture states that every graph of minimum degree at least 3 contains a simple cycle whose length is a power of two. In this paper, we prove that the conjecture holds for Cayley graphs on generalized quaternion groups, dihedral groups, semidihedral groups and groups of order \(p^3\).

Classification of regular balanced Cayley maps of minimal non-abelian metacyclic groups

In this paper, we classify the regular balanced Cayley maps of minimal non-abelian metacyclic groups. Besides the quaternion group Q8, there are two infinite families of such groups which are denoted by Mp, q(m, r) and Mp(n, m), respectively. Firstly, we prove that there are regular balanced Cayley maps of Mp, q(m, r) if and only if q = 2 and we list all of them up to isomorphism. Secondly, we prove that there are regular balanced Cayley maps of Mp(n, m) if and only if p = 2 and n = m or n = m + 1 and there is exactly one such map up to isomorphism in either case. Finally, as a corollary, we prove that any metacyclic p-group for odd prime number p does not have regular balanced Cayley maps.

On the probability of being a deficient square group on 2-element subsets

ABSTRACTTwo elements x and y of a group G satisfy the deficient square property on 2-subsets if |{x,y}2|<4. Let ds(G) be the probability that two randomly chosen elements x and y of G satisfy the deficient square property, that is, xy = yx or . Freiman in 1981 showed that ds(G) = 1 for a finite group G if and only if G is a direct product of the quaternion group of order 8 with an elementary abelian 2-group. We show that if ds(G)<1, then , with equality if and only if G is a direct product of the dihedral group of order 8 with an elementary abelian 2-group.

Hydrodynamics of superfluid quantum space: particle of spin-1/2 in a magnetic field

The modified Navier-Stokes equation describing the velocity field in the superfluid quantum space is loaded by the external Lorentz force introducing electromagnetic fields. In order to open the path for getting the \Schrodinger-Pauli equation describing the behavior of a particle with spin-1/2 in the magnetic field we need to extend the continuity equation to take into account conservation of spin flows on the 3D sphere. This extension includes conservation of the density distribution function in 6D space, that is a multiplication of the 3D Euclidean space by the 3D sphere of unit radius. The special unitary group SU(2) underlies the rotations of the spin on this sphere. This group is isomorphic to the group of quaternions containing the real 4x4 matrices of norm 1. Transition to the quaternion group opens up the way to the possibility of describing the spin-1/2 behavior in a magnetic field as a motion of a spin flag on the 2D sphere. Maxwell's electromagnetic field theory manifests itself in the quaternion group basis by the natural manner.

The Szegö kernel for k-CF functions on the quaternionic Heisenberg group

The tangential k-Cauchy–Fueter operator and the k-CF functions on the quaternionic Heisenberg group are quaternionic counterparts of the tangential CR operator and CR functions on the Heisenberg group in the theory of several complex valuables. We use the group Fourier transform on the quaternionic Heisenberg group to analyze the operator associated the tangential k-Cauchy–Fueter operator and to construct its kernel, from which we get the Szegö kernel of the orthogonal projection from the space of functions to the space of integrable k-CF functions on the quaternionic Heisenberg group.

Propranolol kinetics in plasma from severely burned adults

ObjectivesThe aim of this study was to determine the appropriate propranolol dosing strategy for reducing heart rate in severely burned adults. MethodsA total of 26 patients (≥18 years) with burns covering ≥30% of the total body surface area were included in this IRB-approved study. Plasma propranolol concentrations were determined in a placebo group (n=10) or following one of three dosing strategies: Q6 (n=4), Q8 (n=6), and Q24 (n=6). Blood was collected just before dosing and at regular intervals over two dosing periods with corresponding heart rate and blood pressure recordings. Statistical significance was determined by one-way ANOVA followed by the appropriate post-hoc test. ResultsHeart rate was 86±2 bpm for Q6, 93±3 bpm for Q8, and 90±4 bpm for Q24. The Q8 group had a significantly higher heart rate than the Q6 group (p=0.0001). Plasma propranolol concentrations were significantly higher in the Q6 dosing strategy than in the Q8 dosing strategy (p=0.02). ConclusionsHeart rate can be decreased to a similar degree with Q6 and Q24 dosing strategies, with the Q8 dosing strategy being less effective. Q6 dosing is recommended to maintain reduced heart rate throughout dosing periods.

Smallest graphs with given generalized quaternion automorphism group

AbstractFor , a smallest graph whose automorphism group is isomorphic to the generalized quaternion group is constructed. If , then such a graph has vertices and edges. In the special case when , a smallest graph has 16 vertices but 44 edges.

Structure of irreducibly covariant quantum channels for finite groups

We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation (U) of a finite group (G), whenever U⊗Uc is simply reducible (with Uc being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of any such linear map. The eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group G. This in turn yields necessary and sufficient conditions on the eigenvalues of any such linear map for it to be a quantum channel. We also obtain a wide class of quantum channels which are irreducibly covariant by construction. For two-dimensional irrreducible representations of the symmetric group S(3), and the quaternion group Q, we also characterize quantum channels which are both irreducibly covariant and entanglement breaking.

On a microscopic representation of space–time IV

We summarize some previous work on SU(4) describing hadron representations and transformations as well as its noncompact “counterpart” SU*(4) being the complex embedding of SL(2,H). So after having related the 16-dim Dirac algebra to SU*(4), on the one hand we have access to real, complex, and quaternionic Lie group chains and their respective algebras, on the other hand it is of course possible to relate physical descriptions to the respective representations. With emphasis on the common maximal compact subgroup USp(4), we are led to projective geometry of the real 3-space and various transfer principles which we use to extend the previous work on the rank 3-algebras above. On real spaces, such considerations are governed by the groups SO(n,m) with n + m = 6. The central thread, however, focuses here on line and Complex geometry which finds its well-known counterparts in descriptions of electromagnetism and special relativity as well as—using transfer principles—in Dirac, gauge, and quantum field theory. We discuss a simple picture where Complexe of second grade play the major and dominant rôle to unify (real) projective geometry, complex representation theory and line/Complex representations in order to proceed to dynamics.

On the 2k-step Jordan-Fibonacci sequence

In this paper, we define the 2k-step Jordan-Fibonacci sequence, and then we study the 2k-step Jordan-Fibonacci sequence modulo m. Also, we obtain the cyclic groups from the multiplicative orders of the generating matrix of the 2k-step Jordan-Fibonacci sequence when read modulo m, and we give the relationships among the orders of the cyclic groups obtained and the periods of the 2k-step Jordan-Fibonacci sequence modulo m. Furthermore, we extend the 2k-step Jordan-Fibonacci sequence to groups, and then we examine this sequence in the finite groups. Finally, we obtain the period of the 2k-step Jordan-Fibonacci sequence in the generalized quaternion group Q_{2^{n}} as applications of the results produced.

A note on division algebras of dimension four

In this paper we consider a four-dimensional real algebra [Formula: see text] and prove that if the quaternion group acts on a certain subset of End[Formula: see text] transitively, then [Formula: see text] is a division algebra. We will also show that under certain technical conditions [Formula: see text] has no identity. Using these results we can explicitly construct a 16-parameter family of four-dimensional real division algebras. In addition, we will find a one-parameter family of such algebras with trivial derivation algebras.

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter

It is possible to describe fermionic phases of matter and spin-topological field theories in 2+1d in terms of bosonic “shadow” theories, which are obtained from the original theory by “gauging fermionic parity”. The fermionic/spin theories are recovered from their shadow by a process of fermionic anyon condensation: gauging a one-form symmetry generated by quasi-particles with fermionic statistics. We apply the formalism to theories which admit gapped boundary conditions. We obtain Turaev-Viro-like and Levin-Wen-like constructions of fermionic phases of matter. We describe the group structure of fermionic SPT phases protected by ℤ2f × G. The quaternion group makes a surprise appearance.

On Artin's Character Table and Artin's Cokernel of the Group(Q2m×C4) When m=2h,h∈Z+

The main purpose of this paper is to find Artin's characters table of the group (Q2m×C4)when m=2h, h∈Z+, which is denoted by Ar(Q2m ×C4) where Q2m is denoted to Quaternion group and C4 is the cyclic group of order 4 .Moreover we have found The rational valued characters table and the cyclic decomposition of Artin's cokernel AC(Q2m ×C4) when m=2h, h∈Z+.

On Artin's characters table of the group (Q2m Cp) when m is an odd number and p is prime number

In this paper, we prove that the general form of Artin's characters table of the group (Q2m Cp ) such that&#x0D; Q2m be the Quaternion group of order 4m when m is an odd number and Cp be the cyclic group of order p&#x0D; when p is prime number and (Q2m×Cp) be direct product of Q2m and Cp such that&#x0D; (Q2m Cp ) = {(q,c):q Q2m ,c Cp} and |Q2m×Cp|=|Q2m|.|Cp|=4m.p=4pm.&#x0D; This table which depends on Artin's characters table of a quaternion group of order 4m when m is an odd&#x0D; number. which is denoted by Ar(Q2m Cp ).

On products of groups with abelian subgroups of small index

Abstract It is proved that every group of the form G = A ⁢ B {G=AB} with two subgroups A and B each of which is either abelian or has a quasicyclic subgroup of index 2 is soluble of derived length at most 3. In particular, if A is abelian and B is a locally quaternion group, this gives a positive answer to Question 18.95 of the “Kourovka notebook” posed by A. I. Sozutov.

Some types and covers for quaternionic hermitian groups

ABSTRACTWe determine reducibility points for a certain family of induced representations for quaternionic hermitian and anti-hermitian groups over a p-adic field. We do this via Bushnell and Kutzko’s method of types and covers.

On the second 2-class group [formula omitted] of some imaginary quartic cyclic number field K

Let H8 denote the quaternion group of order 8 and put G=Z/2Z×H8. Let K be some imaginary quartic cyclic number field whose 2-class group is of type (2,2,2). In this paper, we prove in particular that G is realizable over K, i.e., G≃Gal(K2(2)/K) where K2(2) is the second Hilbert 2-class field of K. Then we study the capitulation problem of the 2-ideal classes of K in the fourteen intermediate unramified extensions between K and its first Hilbert 2-class field. Additionally, these fourteen unramified extensions are constructed, and the abelian type invariants of their 2-class groups and the length of the 2-class field tower of K are given.

Functions of the infinitesimal generator of a strongly continuous quaternionic group

The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that [Formula: see text] is the quaternionic infinitesimal generator of a strongly continuous group of operators [Formula: see text] and we show how we can define bounded operators [Formula: see text], where [Formula: see text] belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace–Stieltjes transform. This class includes functions that are slice regular on the [Formula: see text]-spectrum of [Formula: see text] but not necessarily at infinity. Moreover, we establish the relation between [Formula: see text] and the quaternionic functional calculus and we study the problem of finding the inverse of [Formula: see text].

Unit graphs and subgraphs of Symmetric, Quaternion and Heisenberg groups

We represent finite group in the form of a graph, these graphs are called unit graph. In this paper we shall study unit graphs of Symmetric, Quaternion and Heisenberg groups. Also study of different properties like the subgroups of a group, normal subgroups of a group are carried out using the unit graph of the group.

Novel Noncommutative Cryptography Scheme Using Extra Special Group

Noncommutative cryptography (NCC) is truly a fascinating area with great hope of advancing performance and security for high end applications. It provides a high level of safety measures. The basis of this group is established on the hidden subgroup or subfield problem (HSP). The major focus in this manuscript is to establish the cryptographic schemes on the extra special group (ESG). ESG is showing one of the most appropriate noncommutative platforms for the solution of an open problem. The working principle is based on the random polynomials chosen by the communicating parties to secure key exchange, encryption-decryption, and authentication schemes. This group supports Heisenberg, dihedral order, and quaternion group. Further, this is enhanced from the general group elements to equivalent ring elements, known by the monomials generations for the cryptographic schemes. In this regard, special or peculiar matrices show the potential advantages. The projected approach is exclusively based on the typical sparse matrices, and an analysis report is presented fulfilling the central cryptographic requirements. The order of this group is more challenging to assail like length based, automorphism, and brute-force attacks.