Quaternion Group Research Articles

Product of two involutions in quaternionic special linear group

Abstract An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify the reversible and strongly reversible elements in the quaternionic special linear group $ \mathrm {SL}(n,\mathbb {H})$ and quaternionic projective linear group $ \mathrm {PSL}(n,\mathbb {H})$ . We prove that an element of $ \mathrm {SL}(n,\mathbb {H})$ (resp. $ \mathrm {PSL}(n,\mathbb {H})$ ) is reversible if and only if it is a product of two skew-involutions (resp. involutions).

Metric and strong metric dimension in TI-power graphs of finite groups

<p>Given a finite group $ G $ with the identity $ e $, the TI-power graph (trivial intersection power graph) of $ G $ is an undirected graph with vertex set $ G $, in which two distinct vertices $ x $ and $ y $ are adjacent if $ \langle x\rangle\cap \langle y\rangle = \{e\} $. In this paper, we obtain closed formulas for the metric and strong metric dimensions of the TI-power graph of a finite group. As applications, we compute the metric and strong metric dimensions of the TI-power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a semi-dihedral group.</p>

Continuous functions on primal topological spaces induced by group actions

<p>If $ G $ is a group acting on a set $ X $, then for any $ a\in G $, the restriction $ \phi_a:X\to X $ of the action to $ a $ induces a topology $ \tau_a $ for $ X $, called the primal topology induced by $ \phi_a $. First, we obtain a characterization of the normal subgroups in terms of the primal topologies. Later, we prove that some commutative relations among elements on the group $ G $ determine the continuity of maps among different primal spaces $ (X, \tau_{ \phi_x}) $. In particular, we prove the continuity of some maps when $ a, b, q\in G $ satisfy a quantum type relation, $ ba = qab $, as is in the quaternion and Heisenberg groups.</p>

Zero fibers of quaternionic quotient singularities

We propose a generalization of Haiman’s conjecture on the diagonal coinvariant rings of real reflection groups to the context of irreducible quaternionic reflection groups (also known as symplectic reflection groups). For a reflection group W acting on a quaternionic vector space V , by regarding V as a complex vector space, we consider the scheme-theoretic fiber over zero of the quotient map \pi\colon V \to V/W . For W an irreducible reflection group of (quaternionic) rank at least 6 , we show that the ring of functions on this fiber admits a (g+1)^{n} -dimensional quotient arising from an irreducible representation of a symplectic reflection algebra, where g=2N/n , with N the number of reflections in W and n=\mathrm{dim}_{\mathbf{H}}(V) , and we conjecture that this holds in general. We observe that in fact the degree of the zero fiber is precisely g+1 for the rank one groups (corresponding to the Kleinian singularities). In an appendix, we give a proof that three variants of the Coxeter number, including g , are integers.

Topological Indices of Inverse Graph for Generalized Quaternion Group

In this paper, some of the famous and known topological indices for the inverse graph of Generalized quaternion group, including: Hosoya polynomial, Wiener index, hyper-Wiener index, first Zagreb index, second Zagreb index, ABC index, eccentric index, eccentric-connectivity index, total eccentric index, first Zagreb eccentric index, second Zagreb Eccentric index, graph energy index, and Estrada index.

On the number of quaternion and dihedral braces and Hopf–Galois structures

We prove a conjecture of Guarnieri and Vendramin on the number of braces of a given order whose multiplicative group is a generalised quaternion group. At the same time, we give a similar result where the multiplicative group is dihedral. We also enumerate Hopf-Galois structures of abelian type on Galois extensions with generalised quaternion or dihedral Galois group.

THE PRODUCT OF A GENERALIZED QUATERNION GROUP AND A CYCLIC GROUP

Abstract Let $X=GC$ be a group, where C is a cyclic group and G is either a generalized quaternion group or a dihedral group such that $C\cap G=1$ . In this paper, X is characterized and, moreover, a complete classification for $X$ is given, provided that G is a generalized quaternion group and C is core-free.

Metric and strong metric dimension in intersection power graphs of finite groups

Let G be a finite group and let e be its identity element. The intersection power graph of G is the undirected graph with vertex set G, in which two distinct vertices x and y are adjacent if either 〈 x 〉 ∩ 〈 y 〉 ≠ { e } or one of x and y is e. In this paper, we first compute the strong metric dimension of the intersection power graph of a cyclic group, a dihedral group, and a generalized quaternion group. We also characterize all finite groups G whose intersection power graph has strong metric dimension | G | − 2 . Then, we give the sharp upper and lower bounds for the metric dimension of the intersection power graph of a finite group. As applications, we obtain the metric dimension of the intersection power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

Braces with adjoint group of maximal nilpotency class

Braces with adjoint group of maximal nilpotency class

Homotopy classification of knotted defects in ordered media

We give a homotopy classification of the global defects in ordered media and explain it via the example of biaxial nematic liquid crystals, that is, systems where the order parameter space is the quotient of the three-sphere S 3 by the quaternion group Q . As our mathematical model, we consider continuous maps from complements of spatial graphs to the space S 3 / Q modulo a certain equivalence relation and find that the equivalence classes are enumerated by the six subgroups of Q . Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of Q ; once we pass to planar diagrams, these labels can be refined to elements of Q associated with each arc. The same classification scheme applies not only in the case of Q but also to arbitrary groups.

Several topological indices and entropies for certain families of commutative graphs over Quaternion groups

A group graph is a type of graph formed by combining a group, usually a finite group, with a generating set for that group. Group graphs are employed in various mathematical situations, including algebraic and computational group theory. A graph G is known as a commutative graph if the vertex set of G is a group and two elements are adjacent to each other if they are commuting to each other. In this work, we consider the family of commutative graphs over Quaternion groups. The edge partition mappings related to the degree of each vertex of the graph G are computed. Further, we established many results on various kinds of topological indices and entropies by using M-polynomials. The numerical comparison among computed topological indices has been proposed.

Resistance Distance and Kirchhoff Index of Cayley Graphs on Generalized Quaternion Groups

ABSTRACTBased on irreducible representations of generalized quaternion groups, closed‐form formulae of Kirchhoff indices and resistance distances between vertex pairs of Cayley graphs on these groups are given.

Fuchs' problem for endomorphisms of nonabelian groups

Fuchs' problem for endomorphisms of nonabelian groups

EPR Correlations Using Quaternion Spin

We present a statistical simulation replicating the correlation observed in EPR coincidence experiments without needing non-local connectivity. We define spin coherence as a spin attribute that complements polarization by being anti-symmetric and generating helicity. Point particle spin becomes structured with two orthogonal magnetic moments, each with a spin of 12—these moments couple in free flight to create a spin-1 boson. Depending on its orientation in the field, when it encounters a filter, it either decouples into two independent fermion spins of 12, or it remains a boson and precedes without decoupling. The only variable in this study is the angle that orients a spin on the Bloch sphere, first identified in the 1920s. There are no hidden variables. The new features introduced in this work result from changing the spin symmetry from SU(2) to the quaternion group, Q8, which complexifies the Dirac field. The transition from a free-flight boson to a measured fermion causes the observed violation of Bell’s Inequalities and resolves the EPR paradox.

N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 5 SCFTs and quaternionic reflection groups

It was previously noted that for 3d SCFTs with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} ≥ 6 the moduli space has the form of ℂ4r/Γ, where Γ is a complex reflection group, at least following suitable gauging of finite symmetries. Here we argue that this observation can be extended also to 3d SCFTs with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} ≥ 5 SUSY, where Γ is now a quaternionic reflection group. To do this, we study the moduli space of the known 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 5 SCFTs. For Lagrangian cases, the results for the moduli space are further checked using the superconformal index.

Quaternion Spin

We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, Q8, achieved by multiplying one of the gamma matrices by the imaginary number, i. The reason for doing this is to introduce a bivector into the spin algebra, which complexifies the Dirac field. It then separates into two distinct and complementary spaces: one describing polarization and the other coherence. The former describes a 2D structured spin, and the latter its helicity, generated by a unit quaternion.

Sombor Index and Sombor Polynomial of the Noncommuting Graph Associated to Some Finite Groups

Sombor index is a newly developed degree-based topological index which involves the degree of the vertex in a simple connected graph. The Sombor index is known as the square root of the sum of the squared degrees of two adjacent vertices in a graph. Meanwhile, the noncommuting graph associated to a group is a graph where its vertices are the non-central elements of the group and two vertices are adjacent if and only if they do not commute. In this study, a new notion called the Sombor polynomial is introduced. Then, the general formula of the Sombor index and the Sombor polynomial of the noncommuting graph associated to some finite groups are determined by using their definitions and some preliminaries. The groups involved in this research are the dihedral groups, the quasidihedral groups, and the generalized quaternion groups. The results found can help the chemists and biologists to predict the chemical and physical properties of the molecules without involving any laboratory work.

On the Laplacian Energy of an Orbit Graph of Finite Groups

Let β H denote the orbit graph of a finite group H . Let ζ be the set of commuting elements in H with order two. An orbit graph is a simple undirected graph where non-central orbits are represented as vertices in ζ , and two vertices in ζ are connected by an edge if they are conjugate. In this article, we explore the Laplacian energy and signless Laplacian energy of orbit graphs associated with dihedral groups of order $2w$ and quaternion groups of order 2 w .

Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

Abstract We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p {H^{p}} on quaternionic Siegel upper half space for 2 3 < p ≤ 1 {\frac{2}{3}<p\leq 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

Magnetic fields on non-singular 2-step nilpotent Lie groups

Magnetic fields on non-singular 2-step nilpotent Lie groups