Quaternion Group Research Articles

Normal Cayley digraphs of generalized quaternion groups with CI-property

A Cayley digraph Cay(G,S) of a finite group G with respect to a subset S of G, where S does not contain the identity 1 of G, is said to be a CI-digraph, if Cay(G,S)≅Cay(G,T) implies that G has an automorphism mapping S to T. The group G is called a DCI-group or an NDCI-group if all Cayley digraphs or normal Cayley digraphs of G are CI-digraphs. We prove in this paper that a generalized quaternion group Q4n of order 4n is an NDCI-group if and only if n=2 or n is odd. As a result, we show that if Q4n is a DCI-group then n=2 or n is odd-square-free.

Rigidity, counting and equidistribution of quaternionic Cartan chains

We prove an analog of Cartan's theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.

On the Metric Dimension of the Reduced Power Graph of a Finite Group

Let $G$ be a finite group. The reduced power graph of $G$ is the undirected graph whose vertex set is $G$, and two distinct vertices $x$ and $y$ are adjacent if $\langle x \rangle \subset \langle y \rangle$ or $\langle y \rangle \subset \langle x \rangle$. In this paper, we give tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a $\mathcal{P}$-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

On Symmetric Channels and Codes Over the Quaternion Group

In this paper we study symmetric channels and group codes over the quaternion group Q8. We show that, related to these channels, there is the number CQ8, called group-capacity, which is less or equal than the capacity of the channel. Also we show that CQ8 is an upper bound for the rate of any reliable quaternion group code. Finally we show that the group-capacity equals the channel capacity.

On Degenerate 3-(α,δ)-Sasakian Manifolds

AbstractWe propose a new method to construct degenerate 3-(α,δ)-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate 3-(α,δ)-Sasakian manifolds and prove that no non-trivial compact examples exist aswell as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.

The Commutation Matrices of Elements in Kronecker Quaternion Groups

This article discusses the commutation matrix in the Kronecker quaternion group; that is, a non-abelian group whose 32 elements are matrices of 4 × 4 size, with entries in the set of complex numbers. The purpose of this paper is to describe the commutation matrices obtained in relation to the matrices in this group. The commutation matrix is a permutation matrix that associates the relationship between the vec and vec of the transpose matrix. Based on the classification of matrices in the Kronecker quaternion group, there are 16 classification of commutation matrices for the matrices in this group.

Structure of intersection graphs

<p style="text-align: left;" dir="ltr"> Let <em>G</em> be a finite group and let <em>N</em> be a fixed normal subgroup of <em>G</em>. In this paper, a new kind of graph on <em>G</em>, namely the intersection graph is defined and studied. We use <img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /> to denote this graph, with its vertices are all normal subgroups of <em>G</em> and two distinct vertices are adjacent if their intersection in <em>N</em>. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of <img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /> for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups. </p>

On the RO(G)-graded coefficients of Q8 equivariant cohomology

We calculate the RO(G)-graded coefficients of HZ_, the Eilenberg-MacLane spectrum of constant Mackey functor for quaternion group Q8.

Connectivity Indices of Coprime Graph of Generalized Quarternion Group

Generalized quarternion group (Q_(4n)) is a group of order $4n$ that is generated by two elements x and y with the properties x^{2n}=y^4=e and xy=yx^{-1}. The coprime graph of Q_{4n}, denoted by Omega_{Q_{4n}}, is a graph with the vertices are elements of Q_{4n} and the edges are generated by two elements that have coprime order. The first result of this paper presented that Omega_{Q_{4n}} is a tripartite graph for n is odd and Omega_{Q_{4n}} is a star graph for n is even. The second one presented the connectivity indices of Omega_{Q_{4n}}. Connectivity indices of a graph is a research area in mathematics that popularly applied in chemistry. There are six indices that are presented in this paper, those are first Zagreb index, second Zagreb index, Wiener index, hyper-Wiener index, Harary index, and Szeged index.Generalized quaternion group (Q4n) is a group of order 4n that is generated by two elements x and y with the properties x 2n = y 4 = e and xy = yx−1 . The coprime graph of Q4n, denoted by ΩQ4n , is a graph with the vertices are elements of Q4n and the edges are formed by two elements that have coprime order. The first result of this paper presents that ΩQ4n is a tripartite graph for n is an odd prime and ΩQ4n is a star graph for n is a power of 2. The second one presents the connectivity indices of ΩQ4n . Connectivity indices of a graph is a research area in mathematics that popularly applied in chemistry. There are six indices that are presented in this paper, those are first Zagreb index, second Zagreb index, Wiener index, hyper-Wiener index, Harary index, and Szeged index.

Four-band non-Abelian topological insulator and its experimental realization

Very recently, increasing attention has been focused on non-Abelian topological charges, e.g., the quaternion group Q8. Different from Abelian topological band insulators, these systems involve multiple entangled bulk bandgaps and support nontrivial edge states that manifest the non-Abelian topological features. Furthermore, a system with an even or odd number of bands will exhibit a significant difference in non-Abelian topological classification. To date, there has been scant research investigating even-band non-Abelian topological insulators. Here, we both theoretically explore and experimentally realize a four-band PT (inversion and time-reversal) symmetric system, where two new classes of topological charges as well as edge states are comprehensively studied. We illustrate their difference in the four-dimensional (4D) rotation sense on the stereographically projected Clifford tori. We show the evolution of the bulk topology by extending the 1D Hamiltonian onto a 2D plane and provide the accompanying edge state distributions following an analytical method. Our work presents an exhaustive study of four-band non-Abelian topological insulators and paves the way towards other even-band systems.

Modular forms on indefinite orthogonal groups of rank three

We develop a theory of modular forms on the groups SO(3,n+1), n≥3. This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO(3,n+1), there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO(2,n). As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E8 has rational Fourier expansion.

On Reductants of Two Groups

In this paper we consider the reductant of the dihedral group Dn, consisting of a set of axial symmetries, and the sphere S2 as a reductant of the group SU(2,C) ∼= S3 (the group of unit quaternions). By introducing the Sabinin’s multiplication on the reductant of Dn, we get a quasigroup with unit

Symmetries and Geometries of Qubits, and Their Uses

The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.

An exotic presentation of Q28

We introduce a new family of presentations for the quaternion groups and show that for the quaternion group of order 28, one of these presentations has non-standard second homotopy group.

Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2

Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P′(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G=K2,2,2,2, in which specific case P(x)=x31. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (12 in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q8 with the three imaginary quaternions i, j, k as generators.

On the group generated by C, P and T: I 2 = T 2 = P 2 = ITP = −1, with applications to pseudo-scalar mesons

We study faithful representations of the discrete Lorentz symmetry operations of parity P and time reversal T , which involve complex phases when acting on fermions. If the phase of P is a rational multiple of π then P 2n = 1 for some positive integer n and it is shown that, when this is the case, P and T generate a discrete group, a dicyclic group (also known as a generalised quaternion group) which are generalisations of the dihedral groups familiar from crystallography. Charge conjugation C introduces another complex phase and, again assuming rational multiples of π for complex phases, TC generates a cyclic group of order 2m for some positive integer m. There is thus a doubly infinite series of possible finite groups labelled by n and m. Demanding that C commutes with P and T forces n = m = 2 and the group generated by P and T is uniquely determined to be the quaternion group. Neutral pseudo-scalar mesons can be simultaneous C and P eigenstates, T commutes with P and C when acting on fermion bi-linears so neutral pseudo-scalar mesons can also be T eigenstates. The T -parity should therefore be experimentally observable and the CPT theorem dictates that T = CP.

Uncertainty Principle for Gabor Transform on the Quaternionic Heisenberg Group

In this paper, we recall the main results of the Fourier transform on the quaternionic Heisenberg group, then we introduce the notion of the Gabor transform on this group and study some of its properties. Finally, we prove some uncertainty principles associated with this transform.

Counting and equidistribution in quaternionic Heisenberg groups

AbstractWe develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebraAover${\mathbb{Q}}$in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points overAin quaternionic Heisenberg groups.

Properties of the matrix from Kronecker product on the representation of quaternion group

This paper discusses the properties of matrices, which are elements of a group derived from the application of Kronecker product to the representation of the quaternion group (the author calls this group with Kronecker quaternion group). The properties of the new matrices constructed by matrices from the Kronecker quaternion group as submatrix in a partitioned matrix are discussed based on transpose, determinant, and permutation matrix.

Some Special Graphs of Quaternion Group

Research on an algebraic structure represented in graph theory opens the way for new research in recent years. Several types of new graphs continue to be developed, such as coprime and non-coprime graphs. This article will represent the quaternion group in several graphs, such as coprime graphs, non-coprime graphs, commuting graphs, non-commuting graphs, and identity graphs. We obtained several theorems about unique graphs. One of the results is that non-coprime graphs from the quaternion group are complete and regular graphs.