Search

In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]

We compute the ring of non-induced representations for a cyclic group, Cn, over an arbitrary field and show that it has rank φ (n) , where φ is Euler's totient function—independent of the characteristic of the field. Along the way, we obtain a "pick-a-number" trick; expressing an integer n as a sum of products of p-adic digits of related integers. [ABSTRACT FROM AUTHOR]

AbstractLet ℤn be the ring of integers modulo n. Let Ct , Em , and Fr,s respectively denote the cyclic group of order t, the elementary abelian 2-group of order 2m , and the abelian group of exponent 4 with order 2r4s . In this article, we find the structure and generators of the unit group V(ℤnC2). We also solve the normal complement problem in V(ℤnC2) . Additionally, we provide a normal complement of Em in V(ℤ2nEm). At the end, we determine the structure of V(ℤpnFr,s) for an odd prime p and establish that Fr,s does not have a normal complement in V(ℤpnFr,s) . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let G be a finite group and k a fixed positive integer. We define the generalized order divisor graph of G to be a graph whose vertex set is the group G and in which two vertices a and b are adjacent if and only if the orders o(ak) and o(bk) are different and either o(ak) divides o(bk) or o(bk) divides o(ak). This generalizes the order divisor graphs of finite groups. Some properties of our graph are introduced, and we investigate the structure of the generalized order divisor graphs of finite cyclic groups. [ABSTRACT FROM AUTHOR]

K. R. Nagarajan constructed an example of a formal power series ring of dimension two, over a field of characteristic two, with the action of a cyclic group of order two, such that the ring of invariants is not noetherian. We point out how Nagarajan's example readily extends to each positive prime characteristic, and also to a characteristic zero example: There exists a formal power series ring of dimension two, over a field of characteristic zero, with an action of the infinite cyclic group, such that the ring of invariants is not noetherian. Both the positive characteristic and the characteristic zero examples are sharp in multiple ways. [ABSTRACT FROM AUTHOR]

In this paper, the group homomorphism graph is introduced and investigated. The group homomorphism graph, denoted by HI(G,G′), is an undirected graph in which the vertex set contains all homomorphisms excluding the monomorphisms and the zero homomorphism from the group G to the group G′, and two distinct vertices are adjacent if and only if the intersection of their kernels is non-trivial. We investigate the interplay between the graph-theoretic properties of this graph with algebraic properties of the groups. In this work, connectedness, diameter, clique number, chromatic number, domination number, independence number, etc. are found. [ABSTRACT FROM AUTHOR]

The problem of synchronization over a group G aims to estimate a collection of group elements G 1 ⁎ , ... , G n ⁎ ∈ G based on noisy observations of a subset of all pairwise ratios of the form G i ⁎ G j ⁎ − 1. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup — an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error. [ABSTRACT FROM AUTHOR]

Let C n be a cyclic group of order n. A sequence S of length ℓ over C n is a sequence S = a 1 ⋅ a 2 ⋅ ... ⋅ a ℓ of ℓ elements in C n , where a repetition of elements is allowed and their order is disregarded. We say that S is a zero-sum sequence if Σ i = 1 ℓ a i = 0 and that S is a zero-sum free sequence if S contains no zero-sum subsequence. In 2000, Gao obtained a construction of all zero-sum free sequences of length n − 1 − k over C n for 0 ≤ k ≤ n 3 . In this paper, we consider a generalization for a random subset of C n. Let R = R (C n , p) be a random subset of C n obtained by choosing each element in C n independently with probability p. Let N n − 1 − k R be the number of zero-sum free sequences of length n − 1 − k in R. Also, let N n − 1 − k , d R be the number of zero-sum free sequences of length n − 1 − k having d distinct elements in R. We obtain the expectations of N n − 1 − k R and N n − 1 − k , d R for 0 ≤ k ≤ n 3 and show that N n − 1 − k R and N n − 1 − k , d R are asymptotically almost surely (a.a.s.) concentrated around their expectations when k is fixed. Moreover, we provide two ways to compute the expectations using partition numbers and a recurrence formula. [ABSTRACT FROM AUTHOR]

The proper power graph P ∗ (G) of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups G 1 and G 2 , where at least one of G 1 and G 2 is an EPO group. Also we provide the full spectrum and hence the energy of P ∗ (G 1 × G 2) whenever both G 1 and G 2 are EPO groups. [ABSTRACT FROM AUTHOR]

The two key parameters in this study of graph theory are the rainbow connected number and strongly rainbow connected numbers. The purpose of this study is to identify the strongly rainbow connected numbers of the inverse graphs corresponding to the modular groups under various operations, and to evaluate and explain the characteristics of these numbers in various contexts. The objective of this Study is to apply these concepts and results in the field of Networking and Coding. An edge colored graph G is called rainbow-connected if a path whose edges have different colors that connects any two vertices. The minimum k for which there exist a rainbow k-coloring of G is called the rainbow connection number of G, denoted by rc(G), the adjacent edges may be allowed to color with the same color. The graph G is strongly rainbow connected if there exists a rainbow u-v geodesic for every pair of vertices u and v in G. [ABSTRACT FROM AUTHOR]

Let G be an abelian group and A 1 , ... , A k (k ≥ 2) be nonempty subsets of G. The sets A 1 , ... , A k are said to form a complete decomposition of G of order k if G = A 1 + ⋯ + A k and A 1 , ... , A k are pairwise disjoint. The size of a complete decomposition A 1 , ... , A k of G is defined to be | ∪ i = 1 k A i | = ∑ i = 1 k | A i |. In this paper, we determine the minimum and maximum size of a complete decomposition of a finite cyclic group. [ABSTRACT FROM AUTHOR]

An element a in a ring R is said to be of (m, k)-type if am = ak where m and k are positive integers with m > k ≥ 1. Let Xn(m, k) be the set of all (m, k)-type elements, X* n(m, k) be the set of all nonzero (m, k)-type elements, and Sn(m, k) be the set of all nonunit (m, k)-type elements in the ring of integers modulo n. In this paper, we study the algebraic structures of Xn(m, k), X* n(m, k) and Sn(m, k) and characterize all values of n, m, and k for which Xn(m, k) and Sn(m, k) are cyclic semigroups and X* n(m, k) is a cyclic group. [ABSTRACT FROM AUTHOR]

The interplay of groups and graphs has been a subject of interest by mathematics researchers nowadays. One particular instance is the identity graph of a group introduced by Kandasamy [6]. Moreover, the concept of a central graph of any graph is widely used by many graph theorist. The central graph of a graph G denoted by C(G) can be obtained by subdividing the edge of G exactly once and joining all the nonadjacent vertices of G in C(G). In this paper, we construct the central graph of the identity graph of finite cyclic group and investigate some of its graph properties [ABSTRACT FROM AUTHOR]

Infinite cyclic groups created by various objects belong to the class to the class basic algebraic structures. In this paper, we construct the infinite cyclic group of differential neurons which are modifications of artificial neurons in analogy to linear ordinary differential operators of the n-th order. We also describe some of their basic properties. [ABSTRACT FROM AUTHOR]

In this paper, we consider the finite groups H(t,l,m)=⟨a,b,c|at=bl=cm=1,[a,b]=c,[a,c]=[b,c]=1⟩.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} H_{(t,l,m)}=\langle a,b,c | a^t=b^l=c^m=1, [a,b]=c, [a,c]=[b,c]=1\rangle . \end{aligned}$$\end{document}We obtain the generalized order-k Pell sequences and study their periods. We prove the period of the order-k Pell sequence divided the period of the generalized order-k Pell sequence in the Heisenberg group. Then, the generalized order-k Pell sequence in Heisenberg group are used to define new cyclic groups. As an application, these groups are used in encryption algorithms. [ABSTRACT FROM AUTHOR]

Let G be a finite group which is not cyclic of prime power order. The join graph Δ(G) of G is a graph whose vertex set is the set of all proper subgroups of G, which are not contained in the Frattini subgroup G and two distinct vertices H and K are adjacent if and only if G = ⟨H;K⟩. Among other results, we show that if G is a finite cyclic group and H is a finite group such that Δ(G) ≅ Δ(H), then H is cyclic. Also we prove that Δ(G) ≅ Δ(A5) if and only if G ≅ A5. [ABSTRACT FROM AUTHOR]

Monad graph was introduced by V.I. Arnold in 2003 and many result were investigated in. In this paper is we study the properties of monad graph generated by dynamical system defined on finit group which is isomorphic to Cn cyclic group of order n and Dihedral group D2n. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements and G be a finite abelian group. In this work we gave conditions to ensure that a code in F q G is a one-weight code in the case when G is a cyclic group with n elements, such that gcd (n , q) = 1 , and also when G is an abelian group. [ABSTRACT FROM AUTHOR]

This article is concerned with establishing the structure of the unit group of certain group algebras. In particular, we establish the structure of the unit group of the group algebra F2k (Cn×D8) for n ≥ 1. [ABSTRACT FROM AUTHOR]

The power graph 𝒫 (G) of a finite group G is the undirected simple graph whose vertex set is G , in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer n ≥ 2 , let C n denote the cyclic group of order n and let r be the number of distinct prime divisors of n. The minimum degree δ (𝒫 (C n)) of 𝒫 (C n) is known for r ∈ { 1 , 2 } , see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For r ≥ 3 , under certain conditions involving the prime divisors of n , we identify at most r − 1 vertices such that δ (𝒫 (C n)) is equal to the degree of at least one of these vertices. If r = 3 , or that n is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of n. [ABSTRACT FROM AUTHOR]

A linear configuration is said to be common in an Abelian group G if every 2-coloring of G yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked whether, analogously to a result by Jagger, Šťovíček and Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in F p n for p ≥ 5 and large n and in Z p for large primes p. [ABSTRACT FROM AUTHOR]

Recalling each edge of a graph H has 2 oppositely oriented arcs, each vertex v of H is identified with the set of arcs, denoted (v , e) , departing from v along the edges e of H incident to v. Let H be a (λ , μ) -biregular graph with bipartition (Y , X) , where | Y | = k μ and | X | = k λ , (0 < k , λ , μ ∈ Z). We consider the problem, for each edge e = y x in H , of assigning, a color (given by an element) of Y , resp. X , to the arc (y , e) , resp. (x , e) , so that each color is assigned exactly once in the set of arcs departing from each vertex of H. Furthermore, we set such assignment to fulfill a specific bicolor weight function over a monotonic subset of Y × X. This problem applies to the Design of Experiments for Industrial Chemistry, Molecular Biology, Cellular Neuroscience, etc. An algorithmic construction based on biregular graphs with bipartitions given by cyclic-group pairs is presented, as well as 3 essentially different solutions to the Great Circle Challenge Puzzle based on a different biregular graph whose bipartition is formed by the vertices and 5-cycles of the Petersen graph. [ABSTRACT FROM AUTHOR]

An endomorphism h of a group G is said to be strong whenever for every congruence θ on G, (x,y) ∈ θ implies (h(x), h(y)) ∈ θ for every x, y ∈ G. A group G is said to have the strong endomorphism kernel property if every congruence on G is the kernel of a strong endomorphism. In this note, we study the strong endomorphism kernel property in the class of Abelian groups. In particular, we show that a finite Abelian group has the strong endomorphism kernel property if and only if it is cyclic. [ABSTRACT FROM AUTHOR]

Hamiltonicity of graphs possessing symmetry has been a popular subject of research, with focus on vertex-transitive graphs, and in particular on Cayley graphs. In this paper, we consider the Hamiltonicity of another class of graphs with symmetry, namely covering graphs of trees. In particular, we study the problem for covering graphs of trees, where the tree is a voltage graph over a cyclic group. Batagelj and Pisanski were first to obtain such a result, in the special case when the voltage assignment is trivial; in that case, the covering graph is simply a Cartesian product of the tree and a cycle. We consider more complex voltage assignments, and extend the results of Batagelj and Pisanski in two different ways; in these cases the covering graphs cannot be expressed as products. We also provide a linear time algorithm to test whether a given assignment satisfies these conditions. [ABSTRACT FROM AUTHOR]

Let V , W {V,W} be representations of a cyclic group G of prime order p over a field 𝕜 {\Bbbk} of characteristic p. The module of covariants 𝕜 ⁢ [ V , W ] G {\Bbbk[V,W]^{G}} is the set of G-equivariant polynomial maps V → W {V\rightarrow W} , and is a module over 𝕜 ⁢ [ V ] G {\Bbbk[V]^{G}}. We give a formula for the Noether bound β ⁢ (𝕜 ⁢ [ V , W ] G , 𝕜 ⁢ [ V ] G) {\beta(\Bbbk[V,W]^{G},\Bbbk[V]^{G})} , i.e. the minimal degree d such that 𝕜 ⁢ [ V , W ] G {\Bbbk[V,W]^{G}} is generated over 𝕜 ⁢ [ V ] G {\Bbbk[V]^{G}} by elements of degree at most d. [ABSTRACT FROM AUTHOR]

In this paper, we investigate domination number,γ, as well as signed domination number,γS, of all cubic Cayley graphs of cyclic and quaternion groups. In addition, we show that the domination and signed domination numbers of cubic graphs depend on each other. [ABSTRACT FROM AUTHOR]

Let G be a group. The power graph of G is a graph with vertex set G in which two distinct elements x, y are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph on the set G - { 1 } . A group whose proper power graph is connected must be either a torsion group or a torsion-free group; we give characterizations of some groups whose proper power graphs are connected. [ABSTRACT FROM AUTHOR]

In this paper we prove that a permutation group A generated by a single permutation σ is an automorphism group of an edge-colored graph if and only if for every orbit O of σ with cardinality | O | ≥ 3 there exists another orbit O ′ of σ such that gcd (| O | , | O ′ |) ≥ 3. We prove also that every such permutation group is an automorphism group of a 3-colored graph. This is a step towards a characterization of cyclic permutation groups that are automorphism groups of simple graphs. [ABSTRACT FROM AUTHOR]

In this paper, we compute the number of subgroups, normal, cyclicsubgroups, maximal, minimal and Sylow subgroups of the group G =D2n × C2. [ABSTRACT FROM AUTHOR]

The power graph 𝒫 (G) of a given finite group G is the simple undirected graph whose vertices are the elements of G , in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity κ (𝒫 (G)) of 𝒫 (G) is the minimum number of vertices which need to be removed from G so that the induced subgraph of 𝒫 (G) on the remaining vertices is disconnected or has only one vertex. For a positive integer n , let C n be the cyclic group of order n. Suppose that the prime power decomposition of n is given by n = p 1 n 1 p 2 n 2 ⋯ p r n r , where r ≥ 1 , n 1 , n 2 , ... , n r are positive integers and p 1 , p 2 , ... , p r are prime numbers with p 1 < p 2 < ⋯ < p r . The vertex connectivity κ (𝒫 (C n)) of 𝒫 (C n) is known for r ≤ 3 , see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl.17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for r ≥ 4 , we give a new upper bound for κ (𝒫 (C n)) and determine κ (𝒫 (C n)) when n r ≥ 2. We also determine κ (𝒫 (C n)) when n is a product of distinct prime numbers. [ABSTRACT FROM AUTHOR]

Widespread deployment of RFID system arises security and privacy concerns of users. There are several proposals in the literature to avoid these concerns, but most of them provide reasonable privacy at the cost of search complexity on the server side. The search complexity increases linearly with the number of tags in the system. Some schemes use a group based approach to solve the search complexity problem. In this paper, we proposed a group based authentication protocol for RFID system which is based on some characteristics of cyclic groups. The scheme uses only bitwise XOR, circular left shift (≪ ) and modulo operations for the computational work. Also, the scheme does not use any pseudo-random number generator on the tag side. We use two benchmark metric based on anonymity set to measure the privacy level of the system when some tags are compromised by an adversary. We present some simulation results which show that the scheme preserves the high level of privacy and discloses very less amount of information when some tags are compromised. Furthermore, its formal and informal analysis show that our scheme preserves information privacy as well as un-traceability and also withstand various well-known attacks. [ABSTRACT FROM AUTHOR]

Let Z n be the additive group of residue classes modulo n. Let c (n 1 , n 2 , n 3) denote the number of cyclic subgroups of the group Z n 1 × Z n 2 × Z n 3 , where n 1 , n 2 and n 3 are arbitrary positive integers. In this paper we obtain an asymptotic formula for the sum ∑ n 1 , n 2 , n 3 ≤ x c (n 1 , n 2 , n 3). [ABSTRACT FROM AUTHOR]

The paper is a survey type article in which we present some results on projectively equivalent representations of finite groups. [ABSTRACT FROM AUTHOR]

Group Theory is a powerful “tool” which has been extensively exploited to reduce the computational cost in the numerical analysis of several physical problems, including computational electromagnetics. This article proposes the extension of the application of group theory to an integral formulation of the eddy current problems in the presence of n-fold rotational symmetries. The proposed approach is applied to the eddy currents computation in toroidal fusion devices, where rotational symmetries play a key role. [ABSTRACT FROM AUTHOR]

In this paper we establish conditions for a permutation group generated by a single permutation to be an automorphism group of a graph. This solves the so called concrete version of König's problem for the case of cyclic groups. We establish also similar conditions for the symmetry groups of other related structures: digraphs, supergraphs, and boolean functions. [ABSTRACT FROM AUTHOR]

The security of data communicated through wireless networks is a challenging issue due to the presence of malicious and unauthenticated users whose intention is either to disrupt the communication or to know context of the data communicated to perform data theft and prevent the communication made by genuine users. To address this problem, a new secure routing algorithm with a novel encryption scheme is proposed in this paper using the cyclic group based Elliptic curve cryptography in which an encryption key is formed from the points taken from the Beta and Gamma functions in the elliptic key cryptography. For this purpose, the elliptic key is encrypted using the proposed encryption and decryption schemes. Further, it is proved that the proposed model decreases the computational complexity and increases the security, when it is compared to the other signature schemes. This scheme has been tested both analytically and using computer implementation of the proposed secure routing algorithm for wireless sensor networks. It is proved that this proposed scheme is more reliable in providing security than the existing algorithms. The cyclic group used in this work makes the number of elements to be from a finite set of integers and hence the computational complexity is reduced and finite. The experiments have been carried out using Python coding and by transmitting different number of words through the wireless network. It is found that overall complexity for encryption and communication is less in this proposed algorithm than the existing Elliptic Curve Cryptography algorithm. [ABSTRACT FROM AUTHOR]

In this article, various aspects of Laplacian spectra of power graphs of finite cyclic, dicyclic and finite p-groups are considered. The algebraic connectivity is studied and the multiplicity of the Laplacian spectral radius is determined completely for power graphs of all of these groups. Then the equality of the vertex connectivity and the algebraic connectivity is characterized for power graphs of all of the above groups. Orders of dicyclic groups, for which their power graphs are Laplacian integral, are determined. Moreover, it is proved that the notion of equality of the vertex connectivity and the algebraic connectivity and the notion of Laplacian integral are equivalent for power graphs of dicyclic groups. All possible values of Laplacian eigenvalues are obtained for power graphs of finite p-groups. This shows that power graphs of finite p-groups are Laplacian integral. [ABSTRACT FROM AUTHOR]

Let n = p 1 n 1 p 2 n 2 ⋯ p r n r , where r , n 1 , n 2 , ... , n r are positive integers and p 1 , p 2 , ... , p r are distinct prime numbers with p 1 < p 2 < ... < p r. For the finite cyclic group C n of order n , let P (C n) be the power graph of C n and κ (P (C n)) be the vertex connectivity of P (C n). It is known that κ (P (C n)) = p 1 n 1 − 1 if r = 1. For r ≥ 2 , we determine the exact value of κ (P (C n)) when 2 ϕ (p 1 p 2 ⋯ p r − 1) ≥ p 1 p 2 ⋯ p r − 1 , and give an upper bound for κ (P (C n)) when 2 ϕ (p 1 p 2 ⋯ p r − 1) < p 1 p 2 ⋯ p r − 1 , which is sharp for many values of n but equality need not hold always. [ABSTRACT FROM AUTHOR]

We derive a family of prime ideals of the Burnside Tambara functor for a finite group G. In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results towards the same result for a larger class of groups. [ABSTRACT FROM AUTHOR]

Let G be a nontrivial abelian group and let A 1 , ... , A h ( h ≥ 2 ) be nonempty subsets of G. We say that A 1 , ... , A h is a complete decomposition of G of order h if A 1 + ⋯ + A h = G and A i ∩ A j = ∅ for i , j = 1 , ... , h ( i ≠ j ). In this paper we consider the case G is the cyclic group Z n and determine the values of h for which a complete decomposition of Z n of order h exists. The result is then extended to the case G is a finite abelian group. We also investigate the existence of complete decompositions of Z n where the cardinality of each set in the decomposition is a prescribed integer ≥ 2 . As an application, we describe a way to construct codes over a binary alphabet using a construction of a complete decomposition of cyclic groups. [ABSTRACT FROM AUTHOR]

A subset B of an Abelian group G is called a difference basis of G if each element g ∈ G can be written as the difference g = a − b of some elements a , b ∈ B. The smallest cardinality | B | of a difference basis B ⊂ G is called the difference size of G and is denoted by Δ [ G ]. We prove that for every n ∈ ℕ the cyclic group C n of order n has difference size 1 + 4 | n | − 3 2 ≤ Δ [ C n ] ≤ 3 2 n . If n ≥ 9 (and n ≥ 2 ⋅ 1 0 1 5 ), then Δ [ C n ] ≤ 1 2 7 3 n (and Δ [ C n ] < 2 3 n ). Also, we calculate the difference sizes of all cyclic groups of cardinality ≤ 1 0 0. [ABSTRACT FROM AUTHOR]

We revisit the one-dimensional Hubbard model in the case of strong correlations, i.e. for on-site repulsion constant strongly greater than hopping constant u ≫ t , and assume that each atom has only a single relevant orbital. We present in detail the example of half-filled system with four atoms. We reconfirm that when the sites are almost independent atoms, i.e. in the atomic limit, the spin-spin coupling term appears, every site carries exactly one electron and hopping is suppressed. We compare the results with one-dimensional Heisenberg Hamiltonian using the translational and unitary symmetries, and separation of the spin and pseudo-spin degrees of freedom. [ABSTRACT FROM AUTHOR]

The power graph of a group G is the graph whose vertex set is G and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of p -groups are studied. In particular, the number of components of that of abelian p -groups are determined. [ABSTRACT FROM AUTHOR]

We study covering codes of permutations with the $\ell _\infty $ -metric. We provide a general code construction, which combines short building-block codes into a single long code. We focus on cyclic transitive groups as building blocks, determining their exact covering radius, and showing a linear-time algorithm for finding a covering codeword. When used in the general construction, we show that the resulting covering code asymptotically out-performs the best known code while maintaining linear-time decoding. We also bound the covering radius of relabeled cyclic transitive groups under conjugation, showing that the covering radius is quite robust. While relabeling cannot reduce the covering radius by much, the downside is that we prove the covering radius cannot be increased by more than 1 when using relabeling. [ABSTRACT FROM AUTHOR]

Recently, the use of smart cards in human life is increasing. One important aspect in implementing smart cards in human life is its security. Security should be considered because smart cards can contain important data that must be protected, and specific research is needed to address attacks against smart card. There are weaknesses in preserving forward anonymity on dynamic ID-based remote user authentication using smart cards as mentioned in Zhai et al. In their work, attacks that led to the failure of forward anonymity have been successfully carried out. This problem has already been overcome by Lee's scheme, with the probability of the forward anonymity failure was ½296 but the computation cost was still high. This research discussed about overcoming attacks that can lead to the failure of forward anonymity with probability of less than ½296 and has less computation cost. For this purpose, discrete log function and multiplicative inverse is introduced instead of using hash function and Diffie-Hellman method. Based on the discussion it is proven that these functions can preserve the forward anonymity with the probability of failure less than ½296. [ABSTRACT FROM AUTHOR]

In general, for a complex engineering structure with a large number of nodes or members, the inherent symmetry is not easily recognizable. Even though someone succeeds in recognizing certain symmetry properties of the structure, these might be partial ones, and the others will be possibly unnoticed. To overcome this difficulty and enable the integration of computational analysis and symmetry methods, we propose an automated detection method for engineering structures with cyclic symmetries. Only the nodes and the connectivity patterns of the members are needed for implementing this algorithm. Using group theory, we first describe different cyclic groups of symmetries and their symmetry operations. In order to establish a group-theoretic algorithm for automated symmetry detection, several theorems and corollaries are presented. Then, on the basis of matrix representations of symmetry operations, the equivalence of the nodes and members of a structure is evaluated. Hence, the inherent symmetry operations of the structure are identified one by one. Illustrative examples show that the proposed automated symmetry detection method is robust and applicable to both 2D and 3D structures. Highly symmetric structures are recognized accurately and effectively. In addition, asymmetric structures and 2D structures can be recognized in a very small number of iterations. [ABSTRACT FROM AUTHOR]

LetRbe a ring with identity. Then {0} andRare the only additive subgroups ofRif and only ifRis isomorphic (as a ring with identity) to (exactly) one of {0},for a prime numberp. Also, each additive subgroup ofRis a one-sided ideal ofRif and only ifRis isomorphic to (exactly) one of {0},,for an integern≥ 2. This note could find classroom use in a first course on abstract algebra as enrichment material for the unit on ring theory. [ABSTRACT FROM AUTHOR]

The power graphof a groupGis the graph with group elements as vertex set and two elements are adjacent if one is a power of the other. The aim of this paper is to compute the spectrum of the power graph of cyclic groups, dihedral groups, elementary abelian groups of prime power order and the Mathieu group. [ABSTRACT FROM PUBLISHER]