Nontrivial Group Research Articles

Some properties involving feeble compactness, III: (Weakly) compact-bounded topological groups

We study two topological properties weaker than feeble compactness in the class of (para)topological groups, the compact-boundedness and weak compact-boundedness, both introduced by Angoa, Ortiz-Castillo and Tamariz-Mascarúa in [2]. First, given a subgroup H of a topological group G, we show how to extend these properties from the quotient space G/H to G; this, in the cases when H is a compact, locally compact or (weakly) compact-bounded subgroup. Secondly, we prove the main result of this article: if a Tychonoff space X is compact-bounded and not scattered, then the free topological group F(X) and the free Abelian topological group A(X) admit a non-trivial metrizable quotient group; thus extending Theorem 4.7 by Leiderman and Tkachenko in [15]. Finally, we study the r-weakly compact-bounded subsets of a topological space X. We show that r-weak compact-boundedness is a productive property. Moreover, sufficient conditions are given in order for a C-compact subset of a paratopological group G to become an r-weakly compact-bounded subset. This article is part of a larger work developed in [16] and [17].

Planarity of the bipartite graph associated to squares of elements and subgroups of a group

A new kind of bipartite graph is defined in this paper. The work is motivated mainly from a graph introduced by Al-Kaseasbeh and Erfanian [A bipartite graph associated to elements and cosets of subgroups of a finite group, AIMS Math. 6(10) (2021) 10395–10404] and partially from the undirected power graph of a group and from the inclusion graph. For a nontrivial group [Formula: see text], we define a simple undirected bipartite graph [Formula: see text] with the vertex set [Formula: see text], where [Formula: see text] is the set of all elements of the group [Formula: see text] and [Formula: see text] is the set of all subgroups [Formula: see text] of [Formula: see text] such that [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here, we classify all the finite groups [Formula: see text] whose bipartite graphs [Formula: see text] are planar. In addition, we also classify the finite groups [Formula: see text] such that [Formula: see text] is outerplanar, maximal planar, maximal outerplanar, star, tree, claw graph, respectively. The planarity of the graph [Formula: see text] corresponding to an infinite group [Formula: see text] has also been investigated here. It is also observed that [Formula: see text] satisfies Vizing’s conjecture.

The Structure of p-fixed Autonilpotent Finite Groups

The concept of autonilpotent groups was introduced by Moghaddam and Parvaneh in 2010 which is related to concepts of absolute centre and autocommutator subgroups that were first proposed by Hegarty in 1994 and 1997 respectively. In the present article we give some useful properties of such groups. In the first section, we define a subgroup GK_n for a given group G, as GK_n=〈 [g ,α^n ] ┤| g∈ G,α∈Aut(G) 〉 which is characteristic subgroup of G . In this section, based on the induction method, we obtain the structure of GK_n as 〈 [ g ,α ]^n ┤|g∈ G,α∈ Aut(G) 〉, 〈 [ g ,α ]^n [g,α,α]^(n(n-1)/2) ┤|g∈ G,α∈ Aut(G) 〉 and 〈 (∏_(i=0)^(n-1)▒〖[g,α] [g,α,α]^i) [g,α,α,α]^((n(n-1)(n-2))/6) 〗┤|g∈ G,α∈ Aut(G) 〉 in autonilpotent groups of classes 2, 3 and 4 respectively. In second section, n-fixed group is introduced as a group G with GK_n=1 for some n∈N. Then among other results of n-fixed groups, we characterize some n-fixed groups. Based on the role of the absolute centre subgroup in the structure of the group, we prove a non-trivial finite group G is isomorphic to Z_2 iff G is 1-fixed group also we show that abelian group G is p -fixed autonilpotent group iff it is isomorphic to G≅ C_(2^k ) where 1≤ k≤ 3

RIGIDITY FOR VON NEUMANN ALGEBRAS OF GRAPH PRODUCT GROUPS II. SUPERRIGIDITY RESULTS

Abstract In [CDD22], we investigated the structure of $\ast $ -isomorphisms between von Neumann algebras $L(\Gamma )$ associated with graph product groups $\Gamma $ of flower-shaped graphs and property (T) wreath-like product vertex groups, as in [CIOS21]. In this follow-up, we continue the structural study of these algebras by establishing that these graph product groups $\Gamma $ are entirely recognizable from the category of all von Neumann algebras arising from an arbitrary nontrivial graph product group with infinite vertex groups. A sharper $C^*$ -algebraic version of this statement is also obtained. In the process of proving these results, we also extend the main $W^*$ -superrigidity result from [CIOS21] to direct products of property (T) wreath-like product groups.

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

We study an infinite family of j-invariant zero elliptic curves ED:y2=x3+16D and their λ-isogenous curves ED′:y2=x3−27⋅16D, where D and D′=−3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field KD=Q(D) always has non-trivial 3-class group. We prove a series of results related to the set of rational points ED′(Q)∖λ(ED(Q)), and the SL(2,Z)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of ED and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

SUFFICIENT CONDITIONS FOR A GROUP OF HOMEOMORPHISMS OF THE CANTOR SET TO BE TWO-GENERATED

AbstractWe introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is vigorous if for any clopen set A and proper clopen subsets B and C of A, there is $\gamma \in G$ in the pointwise stabiliser of $\mathfrak {C}\backslash A$ with $B\gamma \subseteq C$ . A nontrivial group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is flawless if for all k and w a nontrivial freely reduced product expression on k variables (including inverse symbols), a particular subgroup $w(G)_\circ $ of the verbal subgroup $w(G)$ is the whole group. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of $\operatorname {Homeo}(\mathfrak {C})$ is fairly broad (the class is closed under various natural constructions and contains many well known groups, such as the commutator subgroups of the Higman–Thompson groups $G_{n,r}$ , the Brin-Thompson groups $nV$ , Röver’s group $V(\Gamma )$ , and others of Nekrashevych’s ‘simple groups of dynamical origin’).

Theory of angular momentum transfer from light to molecules

We present a theory describing the interaction of structured light, such as light carrying orbital angular momentum, with molecules. The light-matter interaction Hamiltonian we derive is expressed through couplings between spherical gradients of the electric field and the (transition) electric multipole moments of a particle of any nontrivial rotation point group. Our model can therefore accommodate an arbitrary complexity of the molecular and electric field structure, and it can be straightforwardly extended to atoms or nanostructures. Applying this framework to rovibrational spectroscopy of molecules, we uncover the general mechanism of angular momentum exchange between the spin and orbital angular momenta of light, molecular rotation, and its center-of-mass motion. We show that the nonzero vorticity of Laguerre-Gaussian beams can strongly enhance certain rovibrational transitions that are considered forbidden in the case of nonhelical light. We discuss the experimental requirements for the observation of these forbidden transitions in state-of-the-art spatially resolved spectroscopy measurements. Published by the American Physical Society 2024

Homology groups in CR-warped products of complex space forms

In the present paper, by using the result of Lawson-Simons [15], we adopt a different technique to show that there are no stable integral q-currents and a non-trivial homology group in a compact oriented CR-warped product submanifold Nn in a complex space forms space Qcm(4c) by imposing some restrictions on the squared norm of gradient and the Laplacian of warped function. Finally, we show that the same approach can be given by using Dirichlet energy, positive eigenvalue, and Hamiltonian.

On reversible automata generating lamplighter groups

For any nontrivial abelian group X we construct a reversible automaton such that its set of states and alphabet are identified with X, transition and output functions are defined via the left regular action and its group splits into the restricted wreath product X≀Z, i.e. is a lamplighter group.

Asymptotic teleportation scheme bridging between standard and port-based teleportation

Various modified quantum teleportation schemes are proposed to overcome experimental constraints or to meet specific application requirements for quantum communication. Hence, most schemes are developed and studied with unique methodologies, each with its own inherent challenges. Our research focuses on interconnecting these schemes, which appear to be unrelated to each other, based on the idea that the unique advantages of one scheme can compensate for the limitations of another. In this paper, we introduce an asymptotic teleportation scheme that requires the receiver to complete a classical selection task before performing a quantum correction. This scheme bridges standard teleportation with port-based teleportation through the transformation of joint measurements. Specifically, we categorize and analytically investigate protocols within this scheme for qubit systems. Given that the linear optics teleportation protocol without ancilla qubits is contained in the two non-trivial groups, we provide a novel perspective on its expansion. Furthermore, we discuss the potential application of a protocol from one of these groups as a universal programmable processor and extend these protocols to higher-dimensional systems while maintaining the same properties and potential, providing the analytic form of the joint measurement and its performance. These results thereby propose new avenues for developing a quantum network in higher-dimensional systems.

The quasiprimitive almost elusive groups

Let [Formula: see text] be a nontrivial transitive permutation group on a finite set [Formula: see text] and recall that an element of [Formula: see text] is a derangement if it has no fixed points. Derangements always exist by a classical theorem of Jordan, but there are so-called elusive groups that do not contain any derangements of prime order. In a recent paper, Burness and the author introduced the family of almost elusive groups, which contain a unique conjugacy class of derangements of prime order. In this paper, we complete the classification of the quasiprimitive almost elusive groups.

On threefolds with the smallest nontrivial monodromy group

On threefolds with the smallest nontrivial monodromy group

Groups Generated by Pattern Avoiding Permutations

We study groups generated by sets of pattern avoiding permutations. In the first part of the paper, we prove some general results concerning the structure of such groups. In particular, we consider the sequence ( G n ) n ≥ 0 , where G n is the group generated by a subset of the symmetric group S n consisting of permutations that avoid a given set of patterns. We analyze under which conditions the sequence ( G n ) n ≥ 0 is eventually constant. Moreover, we find a set of patterns such that ( G n ) n ≥ 0 is eventually equal to an assigned symmetric group. Furthermore, we show that any non-trivial simple group cannot be obtained in this way and describe all the non-trivial abelian groups that arise in this way. In the second part of the paper, we carry out a case-by-case analysis of groups generated by permutations avoiding a few short patterns.

Spacetime symmetries and geometric diffusion

We examine relativistic diffusion through the frame and observer bundles associated with a Lorentzian manifold (M, g). Our focus is on spacetimes with a non-trivial isometry group, and we detail the conditions required to find symmetric solutions of the relativistic diffusion equation. Additionally, we analyze the conservation laws associated with the presence of Killing vector fields on (M, g) and their implications for the expressions of the geodesic spray and the vertical Laplacian on both the frame and the observer bundles. Finally, we present several relevant examples of symmetric spacetimes.

A-vertex magicness of product of graphs

In this paper, we discuss the A-vertex magicness of some products of graphs, where A is a non-trivial additive Abelian group with identity 0. We characterize A-vertex magicness of the Cartesian product of Pn and Pm , where A has at least three elements, and we also discuss the A -vertex magicness of co-normal and tensor product of two graphs. Finally, we give a new method to construct an infinite number of A-vertex magic graphs from existing ones using the join operation. As a consequence we obtain an earlier result in [Balamoorthy et al. in AKCE Int. J. Graphs Comb. (2022) 19 (3), 268–275]

Hierarchy of curvatures in exceptional geometry

Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We show that to construct it, a hierarchy of connections is required. They complement the spin connection with higher representations known from the tensor hierarchy. This approach allows to define generalized homogeneous spaces which underlie generalized U-duality, admit consistent truncations and provide a huge class of new flux backgrounds with nontrivial structure groups. Published by the American Physical Society 2024

Flavor invariance of leptonic Yukawa terms in the 3HDM

As an extension of the Standard Model (SM), the 3HDM (Three-Higgs-Doublet Model) defines additional relationships among the fermions. In the visible leptonic Yukawa sector of the 3HDM, we investigate the existence of flavor symmetries under discrete non-abelian groups up to order 1032. When the VEVs are not allowed to depart from their alignment imposed by the condition of minimal potential in the Higgs scalar sector, there will be severe mass degeneracies and a lack of lepton flavor mixing for any nontrivial flavor group. Yet we propose to study the Yukawa terms of the lepton sector for in case the VEV alignment would not be protected, hence leaving the VEV ratios as free parameters. Then, as it turns out, mass splitting for the charged leptons and the neutrinos is obtained although it is still impossible to obtain correct fits to the experimentally known lepton mass spectrum and to the PMNS matrix simultaneously.

Kervaire Conjecture on Weight of Group via Fundamental Group of Ribbon Sphere-Link

Kervaire conjecture that the weight of the free product of every non-trivial group and the infinite cyclic group is not one is affirmatively confirmed by confirming affirmatively Conjecture Z on the knot exterior introduced by Gonzàlez Acuna and Ramırez as a conjecture equivalent to Kervaire conjecture.

Graphs of curves and arcs quasi-isometric to big mapping class groups

Following the works of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinite-type surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically non-trivial big mapping class group to admit such a quasi-isometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe represents the first known example of a hyperbolic mapping class group that is not virtually free.

Nodal phases in non-Hermitian wallpaper crystals

Symmetry and non-Hermiticity play pivotal roles in photonic lattices. While symmetries, such as parity-time (PT) symmetry, have attracted ample attention, more intricate crystalline symmetries have been neglected in comparison. Here, we investigate the impact of the 17 wallpaper space groups of two-dimensional crystals on non-Hermitian band structures. We show that the non-trivial space group representations enforce degeneracies at high symmetry points and dictate their dispersion away from these points. In combination with either T or PT, the symmorphic p4 mm symmetry and the non-symmorphic p2mg, p2gg, and p4gm symmetries protect exceptional chains intersecting at the pertinent high symmetry points.