Quaternion Group Research Articles

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.

Braces with adjoint group of maximal nilpotency class

Computer calculations (Guarnieri and Vendramin, 2017) suggested that the number of braces associated with 2-groups of maximal nilpotency class stabilizes with increasing order. The conjectured seven braces for a generalized quaternion group of order ⩾32 have been classified by the second author (J. Group Theory, 2020). A similar result for dihedral 2-groups was obtained recently by Byott and Ferri. The remaining open case of semidihedral groups is accomplished in the paper.

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

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

Fuchs' problem for endomorphisms of nonabelian groups

In 1960, László Fuchs posed the problem of determining which groups G are realizable as the group of units in some ring R. In [4], we investigated the following variant of Fuchs' problem, for abelian groups: which groups G are realized by a ring R where every group endomorphism of G is induced by a ring endomorphism of R? Such groups are called fully realizable. In this paper, we answer the aforementioned question for several families of nonabelian groups: symmetric, dihedral, quaternion, alternating, and simple groups; almost cyclic p-groups; and groups whose Sylow 2-subgroup is either cyclic or normal and abelian. We construct three infinite families of fully realizable nonabelian groups using iterated semidirect products.

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.

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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

This work has a twofold purpose - the existence study of closed 2-forms, known as magnetic fields, on 2-step nilpotent Lie groups and generating examples for the non-singular family. At the Lie algebra level, while the existence of closed 2-forms for which the center is either nondegenerate or in the kernel of the 2-form, is always guaranteed, the existence of closed 2-forms for which the center is isotropic but not in the kernel of the 2-form, is a special situation. These 2-forms are called of type II. We obtain a strong obstruction for the existence on non-singular Lie algebras. Moreover, we prove that the only H-type Lie groups admitting left-invariant closed 2-forms of type II are the real, complex and quaternionic Heisenberg Lie groups of dimension three, six and seven, respectively. We also prove the non-existence of uniform magnetic fields under certain hypotheses. Finally we give a construction of non-singular Lie algebras, proving that in some families of these examples there are no closed 2-form of type II.

On forbidden subgraphs of main supergraphs of groups

<p>In this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. This is denoted as either $ o(a)\mid o(b) $ or $ o(b)\mid o(a) $, where $ o(\cdot) $ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $ G $ with $ \mathcal{S}(G) $, a cograph that includes when $ G $ is a direct product of two non-trivial groups, as well as when $ G $ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.</p>

Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

<p>In this paper, we introduce a method for computing the primitive decomposition of idempotents in any semisimple finite group algebra, utilizing its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $ \mathbb{C}[D_{2n}] $ and generalized quaternion group algebras $ \mathbb{C}[Q_{4m}] $. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $ \mathbb{C}[D_{8}] $ and $ \mathbb{C}[Q_{8}] $ is described, under which the two complete sets of primitive orthogonal idempotents of these group algebras correspond bijectively.</p>

Spectral Power Graph of Generalized Quaternion Groups

Spectral Power Graph of Generalized Quaternion Groups

Commutativity degree of cyclic chains of some of finite groups

For a finite group G, we introduce the concept of commutativity degree of its cyclic chains. We denote the commutativity degree of cyclic chains of a group G by ccd(G), and compute this measure for some famous groups such as dihedral groups D2n, quaternion groups Q2n, quasi-dihedral groups QD2n, and modular groups Mpn.

QGD-Net: A Lightweight Model Utilizing Pixels of Affinity in Feature Layer for Dermoscopic Lesion Segmentation.

Pixels with location affinity, which can be also called "pixels of affinity," have similar semantic information. Group convolution and dilated convolution can utilize them to improve the capability of the model. However, for group convolution, it does not utilize pixels of affinity between layers. For dilated convolution, after multiple convolutions with the same dilated rate, the pixels utilized within each layer do not possess location affinity with each other. To solve the problem of group convolution, our proposed quaternion group convolution uses the quaternion convolution, which promotes the communication between to promote utilizing pixels of affinity between channels. In quaternion group convolution, the feature layers are divided into 4 layers per group, ensuring the quaternion convolution can be performed. To solve the problem of dilated convolution, we propose the quaternion sawtooth wave-like dilated convolutions module (QS module). QS module utilizes quaternion convolution with sawtooth wave-like dilated rates to effectively leverage the pixels that share the location affinity both between and within layers. This allows for an expanded receptive field, ultimately enhancing the performance of the model. In particular, we perform our quaternion group convolution in QS module to design the quaternion group dilated neutral network (QGD-Net). Extensive experiments on Dermoscopic Lesion Segmentation based on ISIC 2016 and ISIC 2017 indicate that our method has significantly reduced the model parameters and highly promoted the precision of the model in Dermoscopic Lesion Segmentation. And our method also shows generalizability in retinal vessel segmentation.

Degree and distance based topological descriptors of power graphs of finite non-abelian groups

A topological descriptor is a numerical value derived from the molecular structure that encapsulates the most important structural characteristics of the molecule under consideration. Fundamentally, it involves assigning an algebraic value to the composition of chemicals while developing a relationship between this value and several physical properties, like biological activity, and chemical reactivity. This article examines multiple kinds of degree and eccentricity-based topological indices for power graphs of various finite groups. We calculate the Wiener index and its reciprocal, atom-bond connectivity index and its fourth version, the Schultz index, the geometric–arithmetic and harmonic indices, and finally determine the general Randić and Harary indices of power graphs of finite cyclic and non-cyclic groups of order pq, dihedral, and generalized quaternion groups, where p,q(q≥p) are distinct primes.

Generalization of Randic ́ Index of the Non-commuting Graph for Some Finite Groups

Randić index is one of the classical graph-based molecular structure descriptors in the field of mathematical chemistry. The Randić index of a graph is calculated by summing the reciprocals of the square root of the product of the degrees of two adjacent vertices in the graph. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only if they do not commute. In this paper, the general formula of the Randić index of the non-commuting graph associated to three types of finite groups are presented. The groups involved are the dihedral groups, the generalized quaternion groups, and the quasi-dihedral groups. Some examples of the Randić index of the non-commuting graph related to a certain order of these groups are also given based on the main results.

Floquet non-Abelian topological insulator and multifold bulk-edge correspondence

Topological phases characterized by non-Abelian charges are beyond the scope of the paradigmatic tenfold way and have gained increasing attention recently. Here we investigate topological insulators with multiple tangled gaps in Floquet settings and identify uncharted Floquet non-Abelian topological insulators without any static or Abelian analog. We demonstrate that the bulk-edge correspondence is multifold and follows the multiplication rule of the quaternion group Q8. The same quaternion charge corresponds to several distinct edge-state configurations that are fully determined by phase-band singularities of the time evolution. In the anomalous non-Abelian phase, edge states appear in all bandgaps despite trivial quaternion charge. Furthermore, we uncover an exotic swap effect—the emergence of interface modes with swapped driving, which is a signature of the non-Abelian dynamics and absent in Floquet Abelian systems. Our work, for the first time, presents Floquet topological insulators characterized by non-Abelian charges and opens up exciting possibilities for exploring the rich and uncharted territory of non-equilibrium topological phases.

On groups with chordal power graph, including a classification in the case of finite simple groups

AbstractWe prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by (Electron J Combin 28(3):14, 2021). Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic conditions. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups, and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.