Finite Cyclic Group Research Articles

Recalibrating R$\mathbb {R}$‐order trees and Homeo+(S1)$\mbox{Homeo}_+(S^1)$‐representations of link groups

AbstractIn this paper, we study the left‐orderability of 3‐manifold groups using an enhancement, called recalibration, of Calegari and Dunfield's ‘flipping’ construction, used for modifying ‐representations of the fundamental groups of closed 3‐manifolds. The added flexibility accorded by recalibration allows us to produce ‐representations of hyperbolic link exteriors so that a chosen element in the peripheral subgroup is sent to any given rational rotation. We apply these representations to show that the branched covers of families of links associated to epimorphisms of the link group onto a finite cyclic group are left‐orderable. This applies, for instance, to fibred hyperbolic strongly quasi‐positive links. Our result on the orderability of branched covers implies that the degeneracy locus of any pseudo‐Anosov flow on an alternating knot complement must be meridional, which generalises the known result that the fractional Dehn twist coefficient of any hyperbolic fibred alternating knot is zero. Applications of these representations to order detection of slopes are also discussed in the paper.

The universal zero-sum invariant and weighted zero-sum for infinite abelian groups

Let G be an abelian group, and let F ( G ) be the free commutative monoid with basis G. For Ω ⊂ F ( G ) , define the universal zero-sum invariant d Ω ( G ) to be the smallest integer l such that every sequence T over G of length l has a subsequence in Ω . The invariant d Ω ( G ) unifies many classical zero-sum invariants. Let B ( G ) be the submonoid of F ( G ) consisting of all zero-sum sequences over G, and let A ( G ) be the set consisting of all minimal zero-sum sequences over G. The empty sequence, which is the identity of B ( G ) , is denoted by ε . The well-known Davenport constant D ( G ) of the group G can be also represented as d B ( G ) ∖ { ε } ( G ) or d A ( G ) ( G ) in terms of the universal zero-sum invariant. Notice that A ( G ) is the unique minimal generating set of the monoid B ( G ) from the point of view of Algebra. Hence, it would be interesting to determine whether A ( G ) is minimal to represent the Davenport constant or not for a general finite abelian group G. In this paper, we show that except for a few special classes of groups, there always exists a proper subset Ω of A ( G ) such that d Ω ( G ) = D ( G ) . Furthermore, in the setting of finite cyclic groups, we discuss the distributions of all minimal sets by determining their intersections. By connecting the universal zero-sum invariant with weights, we make a study of zero-sum problems in the setting of infinite abelian groups. The universal zero-sum invariant d Ω ; Ψ ( G ) with weights set Ψ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant D Ψ ( G ) (being an special form of the universal invariant with weights) is also investigated for infinite abelian groups. Among other results, we obtain the necessary and sufficient conditions such that D Ψ ( G ) < ∞ in terms of the weights set Ψ when | Ψ | is finite. In doing this, by using the Neumann Theorem on Cover Theory for groups we establish a connection between the existence of a finite cover of an abelian group G by cosets of some given subgroups of G, and the finiteness of weighted Davenport constant.

On Knots with Cyclic Symmetries

This review paper investigates the relationship between symmetries of knots and links in the three-dimensional sphere, with a focus on cyclic symmetries, and their associated polynomial invariants. It examines the behavior of these polynomials in the case where the knot is set-wise fixed by the action of finite cyclic group on the 3-sphere. The study highlights how these invariants can obstruct the existence of such symmetries, providing valuable insights into the topological properties of these knots. The paper reviews established results on polynomial criteria for periodicity and their extensions to freely periodic links. It also explores generalizations to study the symmetries of virtual knots and spatial graphs.

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let M = ( M , ω ) M=(M,\omega ) be either S 2 × S 2 S^2 \times S^2 or C P 2 # C P 2 ¯ \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form ω \omega . Suppose a finite cyclic group Z n \mathbb {Z}_n is acting effectively on ( M , ω ) (M,\omega ) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z n ↪ H a m ( M , ω ) \mathbb {Z}_n\hookrightarrow Ham(M,\omega ) . In this paper, we investigate the homotopy type of the group S y m p Z n ( M , ω ) Symp^{\mathbb {Z}_n}(M,\omega ) of equivariant symplectomorphisms. We prove that for some infinite families of Z n \mathbb {Z}_n actions satisfying certain inequalities involving the order n n and the symplectic cohomology class [ ω ] [\omega ] , the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J J -holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4 4 -manifolds, and on the Chen-Wilczyński classification of smooth Z n \mathbb {Z}_n -actions on Hirzebruch surfaces.

On Nielsen realization and manifold models for classifying spaces

We consider the problem of whether, for a given virtually torsionfree discrete group Γ \Gamma , there exists a cocompact proper topological Γ \Gamma -manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that Γ \Gamma contains a normal torsionfree subgroup π \pi such that π \pi is hyperbolic and π \pi is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and Γ / π \Gamma /\pi is a finite cyclic group of odd order.

Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

Quantum Annular Homology and Bigger Burnside Categories

As part of their construction of the Khovanov spectrum, Lawson, Lipshitz and Sarkar assigned to each cube in the Burnside category of finite sets and finite correspondences, a finite cellular spectrum. In this paper, we extend this assignment to cubes in Burnside categories of infinite sets. This is later applied to the work of Akhmechet, Krushkal and Willis on the quantum annular Khovanov spectrum with an action of a finite cyclic group: we obtain a quantum annular Khovanov spectrum with an action of the infinite cyclic group.

Divisor graphs and symmetries of finite abelian groups

The algebraic composition of an element [Formula: see text] of a commutative binary structure [Formula: see text] is modeled by its divisor (pseudo)graph [Formula: see text] (which can be regarded as a simple graph if [Formula: see text] is an abelian group), whose vertices are the elements of [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. With respect to the action of a group of symmetries on the set of divisor pseudographs of [Formula: see text], the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when [Formula: see text] is a finite abelian group. The minimum number of orbits is computed for every abelian group of order at most 11, and it is also determined for all finite cyclic groups. Moreover, systems of pseudographs that can be realized as those of finite abelian groups are completely characterized. In fact, recurrence relations are given for constructing systems of divisor pseudographs of finite cyclic groups, and all commutative binary structures that are isotopic to a finite abelian group are established.

A Note on the Laplacian Energy of the Power Graph of a Finite Cyclic Group

In this study, the Laplacian matrix concept for the power graph of a finite cyclic group is redefined by considering the block matrix structure. Then, with the help of the eigenvalues of the Laplacian matrix in question, the concept of Laplacian energy for the power graphs of finite cyclic groups was defined and introduced into the literature. In addition, boundary studies were carried out for the Laplacian energy in question using the concepts the trace of a matrix, the Cauchy-Schwarz inequality, the relationship between the arithmetic mean and geometric mean, and determinant. Later, various results were obtained for the Laplacian energy in question for cases where the order of a cyclic group is the positive integer power of a prime.

Gabor frames on finite non-cyclic Abelian groups

We attempt to analyze Gabor frames for L2 spaces on finite non cyclic abelian groups from their natural perspectives and establish their equivalence with Gabor frames on the isomorphic product groups of finite cyclic groups by means of unitary and non unitary invertible linear transformations. Gabor frames and their canonical dual frames on finite non cyclic groups are identified as the images of the same Gabor frame on a finite product of finite cyclic groups under invertible linear transformations.

On the structure of finite groups determined by the arithmetic and geometric means of element orders

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of p ′ -order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors.

On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity

The 3-cyclic refined neutrosophic roots of unity are exactly the solutions of the Diophantine equation in the 3-cyclic refined neutrosophic ring of integers . This paper is dedicated to finding all 3-cyclic refined neutrosophic Diophantine roots of unity, where it proves that there exist only three solutions for the case of odd order (n), and twelve different solutions for the case of even order (n). On the other hand, the group generated from all solutions will be classified as a finite abelian group with direct products of finite cyclic groups.

On The Algebraic Classification of the 4-Cyclic Refined Neutrosophic Real Roots of Unity Group

This paper is dedicated to finding all 4-cyclic refined neutrosophic real solutions of the equation 𝑋𝑛=1 which are called 4-cyclic refined real roots of unity. Also, we classify the algebraic group of these solutions as a direct product of some familiar finite cyclic groups. On the other hand, we illustrate many examples to clarify the validity of our work.

Some Bounds for The Energies of The Power Graphs of Cyclic Groups

In this study, some lower and upper bounds were obtained for the energies of the power graphs of finite cyclic groups by considering the adjacency matrix structure of the power graph of a finite cyclic group. Then, some results are given using the relationship between the case where the order of a cyclic group is the positive integer power of a prime number and the completeness of the power graph corresponding to this cyclic 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.

Artificial Neural Networks Using Quiver Representations of Finite Cyclic Groups

Artificial Neural Networks Using Quiver Representations of Finite Cyclic Groups

Ultra-quantum coherent states in a single finite quantum system

A set of n coherent states is introduced in a quantum system with d-dimensional Hilbert space H(d). It is shown that they resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits. A n-tuple representation of arbitrary states in H(d), analogous to the Bargmann representation, is defined. There are two other important properties of these coherent states which make them ‘ultra-quantum’. The first property is related to the Grothendieck formalism which studies the ‘edge’ of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form that uses vectors in the unit ball of the Hilbert space. It shows that if , the corresponding might take values greater than 1, up to the complex Grothendieck constant . related to these coherent states is shown to take values in the ‘Grothendieck region’ , which is classically forbidden in the sense that does not take values in it. The second property complements this, showing that these coherent states violate logical Bell-like inequalities (which for a single quantum system are quantum versions of the Frechet probabilistic inequalities). In this sense also, our coherent states are deep into the quantum region.

Quotients of skew morphisms of cyclic groups

A skew morphism of a finite group B is a permutation φ of B that preserves the identity element of B and has the property that for every a ∈ B there exists a positive integer ia such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|, called the quotient of φ. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to 161. During the preparation of this paper we noticed a few flaws in Section 5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68–100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.

Some Results on Neutrosophic Group

We present the concept of a neutrosophic group without the use of an indeterminate element “I” in this paper. We also present a similar application to the fundamentals of group theory. We define and study the so-called level subgroups of a neutrosophic subgroup in order to characterize neutrosophic subgroups of finite cyclic groups in a similar way.

Finite cyclic group of p-power order and its compatibility conditions

Finite cyclic groups of p-power order, where p represents a prime number, have long been an interesting field of study in abstract algebra. This paper investigates the compatibility conditions that control their existence and behaviour. An overview of cyclic groups, automorphisms and their properties is given as groundwork for this research. By analysing the interaction between the group's order and its generator, we discovered the compatibility conditions and presented them as the primary finding in this paper.