Finite Nonabelian Group Research Articles

The generalized order (k,t)-Mersenne sequences in groups

The purpose of this paper is to determine the algebraic properties of finite groups via a Mersenne-like sequence. Firstly, we introduce the generalized order $(k,t)$-Mersenne number sequences and study the periods of these sequences modulo $m$. Then, we get some interesting structural results. Furthermore, we expand the generalized order $(k,t)$-Mersenne number sequences to groups and we give the definition of the generalized order $(k,t)$-Mersenne sequences, $MQ_k^t(G,X)$, in the $j$-generator groups and also, investigate these sequences in the non-Abelian finite groups in detail. At last, we obtain the periods of the generalized order $(k,t)$-Mersenne sequences in some special groups as applications of the results produced.

Almost gauge-invariant states and the ground state of Yang-Mills theory

We consider the problem of the explicit description of the gauge-invariant subspace of pure lattice gauge theories in the Hamiltonian formulation, where the gauge group is either a compact Lie group or a finite group. The latter case is particularly interesting for quantum simulation. A basis of states where configurations are grouped according to their holonomies is shown to have several advantages over other descriptions. Using this basis, we compute some properties of interest for some non-Abelian finite groups on small lattices, and in particular we examine the question of whether a certain ansatz introduced long ago is a good approximation for the ground state. Published by the American Physical Society 2024

Superpower graphs of finite groups

For a finite group [Formula: see text] the superpower graph [Formula: see text] of [Formula: see text] is an undirected simple graph with vertex set [Formula: see text] and two vertices are adjacent in [Formula: see text] if and only if the order of one divides the order of the other in [Formula: see text]. The aim of this paper is to provide tight bounds for the vertex connectivity, discuss Hamiltonian-like properties of superpower graph of finite non-Abelian groups having an element of exponent order. We also give some general results about superpower graphs and their relation to other graphs such as the Gruenberg–Kegel graph.

Projective non-commuting graph of a group

Let $G$ be a finite non-abelian group and let $T$ be a transversal of the center of $G$ in $G$. The non-commuting graph of $G$ on a transversal of the center is the graph whose vertices are the non-central elements of $T$ and two vertices $x$ and $y$ are joined by an edge whenever $xy \neq yx$. In this paper, we classify the groups whose non-commuting graph on a transversal of the center is projective.

Double-toroidal and triple-toroidal commuting graph

In this paper, all finite non-abelian groups whose commuting graphs can be embed on the double-torus or triple-torus are classified.

Генерическая NP-полнота проблем разрешимости систем уравнений над конечными группами, полугруппами и полями

In this paper we prove that problems of solvability of systems of equations over finite fields, non-Abelian finite groups and non-commutative finite monoids are complete with respect to generic polynomial reducibility in generic analogue of the NP class.

CONJUGATE S3-MAGIC GRAPHS

In this paper, we introduce conjugate A-magic labeling of graphs where A is a finite non-abelian group and investigate the graphs that are conjugate S3magic.

Universal adjacency spectrum of the commuting and non-commuting graphs on AC-groups

For a simple undirected graph [Formula: see text] and [Formula: see text], [Formula: see text], the universal adjacency matrix [Formula: see text], where [Formula: see text] is the identity matrix, [Formula: see text] is the all-ones matrix, and [Formula: see text] and [Formula: see text] are the adjacency and degree diagonal matrices of [Formula: see text], respectively. For a finite non-abelian group [Formula: see text] and [Formula: see text], the commuting graph [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and the adjacency rule is commuting. The non-commuting graph of [Formula: see text] is the complement of [Formula: see text]. An AC-group is a non-abelian group such that the centralizer of every non-central element is abelian. In this paper, we obtain the universal adjacency eigenpairs of the commuting and non-commuting graphs for an arbitrary AC-group [Formula: see text] with [Formula: see text] or [Formula: see text]. Moreover, we determine the eigenpairs of [Formula: see text] when [Formula: see text] is a specific kind of group.

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.

Method of Security Improvement for MST3 Cryptosystem Based on Automorphism Group of Ree Function Field


 
 
 This article is a part of a research endeavor focused on creating a quantum-resistant cryptosystem for secure encryption and decryption. Our approach employs a challenging word problem while emphasizing cost-effective implementation. Previous research has involved the development of encryption schemes based on high-order groups, offering potential security enhancements. The choice of the non-abelian group is a critical factor in shaping the encryption algorithms, feasibility of implementation, and system parameters. Our central objective is to design a cryptosystem that effectively thwarts quantum cryptanalysis. To achieve this, we employ a logarithmic signature along with a random cover across an entire finite non-abelian group. Our unique contribution lies in optimizing finite group selection, parameters, and circuit solutions for the logarithmic signature to meet specific security and implementation criteria. Within this paper, we introduce an encryption scheme utilizing automorphisms of the Ree functional field and propose a method for enhancing resistance to cryptanalysis through the binding of session keys.
 
 

Symmetries for the 4HDM: extensions of cyclic groups

Multi-Higgs models equipped with global symmetry groups, either exact or softly broken, offer a rich framework for constructions beyond the Standard Model and lead to remarkable phenomenological consequences. Knowing all the symmetry options within each class of models can guide its phenomenological exploration, as confirmed by the vast literature on the two- and three-Higgs-doublet models. Here, we begin a systematic study of finite non-abelian symmetry groups which can be imposed on the scalar sector of the four-Higgs-doublet model (4HDM) without leading to accidental symmetries. In this work, we derive the full list of such non-abelian groups available in the 4HDM that can be constructed as extensions of cyclic groups by their automorphism groups. This list is remarkably restricted but it contains cases which have not been previously studied. Since the methods we develop may prove useful for other classes of models, we present them in a pedagogical manner.

The commuting conjugacy class graphs of finite groups with a given property

Abstract Let G be a finite non-abelian group. The commuting conjugacy class graph Γ ⁢ ( G ) {\Gamma(G)} is defined as a graph whose vertices are non-central conjugacy classes of G and two distinct vertices X and Y in Γ ⁢ ( G ) {\Gamma(G)} are connected by an edge if there exist elements x ∈ X {x\in X} and y ∈ Y {y\in Y} such that x ⁢ y = y ⁢ x {xy=yx} . In this paper, the structure of the commuting conjugacy class graph of group G with the property that G Z ⁢ ( G ) {\frac{G}{Z(G)}} is isomorphic to a Frobenius group of order pq or p 2 ⁢ q {p^{2}q} , is determined.

Plateaued functions on finite nonabelian groups

Plateaued functions on finite nonabelian groups

Nonlinear irreducible characters of finite nonabelian groups

Nonlinear irreducible characters of finite nonabelian groups

Several Zagreb indices of power graphs of finite non-abelian groups

Molecular topology can be described by using topological indices. These are quantitative measures of the essential structural features of a proposed molecule calculated from its molecular structure. It is a numerical value obtained from a molecular configuration that reflects the significant physical characteristics of the suggested molecule. Numerous physical properties, chemical reactivity, and biological activity are correlated with the chemical composition using an algebraic number. The power graph P(G) of a finite group G is a graph whose vertex set is G and in which two distinct vertices are connected by an edge when one element is an integral power of the other. This article investigates a wide range of degree-based topological descriptors for power graphs of various finite groups. We find numerous Zagreb indices (given in Table 1) of power graphs of finite non-cyclic and cyclic groups, dihedral, and generalized quaternion groups.

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case.

An Explicit Minorant for the Amenability Constant of the Fourier Algebra

Abstract We show that if a locally compact group $G$ is non-abelian, then the amenability constant of its Fourier algebra is $\geq 3/2$, extending a result of [9] who proved that this holds for finite non-abelian groups. Our lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by [16]. To do this, we study a minorant for the amenability constant, related to the anti-diagonal in $G\times G$, which was implicitly used in Runde’s work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when $G$ is a countable virtually abelian group, in terms of the Plancherel measure for $G$. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value and present some examples to support the conjecture that the minorant always coincides with the amenability constant.

The cyclic sequences in non-Abelian groups

In this paper, we describe the cyclic 3-nacci sequence and the cyclic generalized order-3 Pell sequence in the non-abelian groups and then we investigate them in non-abelian finite groups in detail. Furthermore, we give the lengths of the periods of the cyclic 3-nacci sequence and the cyclic generalized order-3 Pell sequence in the polyhedral group with respect to the generating triple as applications of the results produced.

On Automorphic Forms of Small Weight for Fake Projective Planes

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the present note we give two short proofs of this statement and show acyclicity of some other line bundles on the fake projective planes with at least $9$ automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The first proof is based on the observation that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite, \emph{non-abelian}, Heisenberg group. For the second proof, we also demonstrate vanishing of odd Betti numbers for a class of abelian covers, and use linearization of an auxiliary line bundle as well.

The Tribonacci-type balancing numbers and their applications

N this paper, we define the Tribonacci-type balancing numbers via a Diophantine equation with a complex variable and then give their miscellaneous properties. Also, we study the Tribonacci-type balancing sequence modulo m and then obtain some interesting results concerning the periods of the Tribonacci-type balancing sequences for any m. Furthermore, we produce the cyclic groups using the multiplicative orders of the generating matrices of the Tribonacci-type balancing numbers when read modulo m. Then give the connections between the periods of the Tribonacci-type balancing sequences modulo m and the orders of the cyclic groups produced. Finally, we expand the Tribonacci-type balancing sequences to groups and give the definition of the Tribonacci-type balancing sequences in the 3-generator groups and also, investigate these sequences in the non-abelian finite groups in detail. In addition, we obtain the periods of the Tribonacci-type balancing sequences in the polyhedral groups (2, 2, n), (2, n, 2), (n, 2, 2), (2, 3, 3), (2, 3, 4), (2, 3, 5).