Finite Abelian Group Research Articles

Co-prime order graphs of finite Abelian groups and dihedral groups

The \textbf{Co-Prime Order Graph} $\Theta (G)$ of a given finite group is a simple undirected graph whose vertex set is the group $G$ itself, and any two vertexes x,y in $\Theta (G)$ are adjacent if and only if $gcd(o(x),o(y))=1$ or prime. In this paper, we find a precise formula to count the degree of a vertex in the Co-Prime Order graph of a finite abelian group or Dihedral group $D_n$.We also investigate the Laplacian spectrum of the Co-Prime Order Graph $\Theta (G)$ when G is finite abelian p-group, ${\mathbb{Z}_p}^t \times {\mathbb{Z}_q}^s$ or Dihedral group $D_{p^n}$. Key Words and Phrases: Co-Prime Order graph,finite abelian group,Dihedral group, Laplacian spectrum.

The large-N limit of the 4d mathcal{N} = 1 superconformal index

We systematically analyze the large-N limit of the superconformal index of mathcal{N} = 1 superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate using the recently-developed technique of elliptic extension. This technique allows us to easily evaluate the integral as a sum over saddle points of an effective action in the limit where the rank of the gauge group is infinite. For a generic quiver theory under consideration, we find a special family of saddles whose effective action takes a universal form controlled by the anomaly coefficients of the theory. This family includes the known supersymmetric black hole solution in the holographically dual AdS5 theories. We then analyze the index refined by turning on flavor chemical potentials. We show that, for a certain range of chemical potentials, the effective action again takes a universal cubic form that is controlled by the anomaly coefficients of the theory. Finally, we present a large class of solutions to the saddle-point equations which are labelled by group homomorphisms of finite abelian groups of order N into the torus.

Minimal cut-sets in the power graphs of certain finite non-cyclic groups

We study minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups. As a result, we find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes.

Construction of Locally Compact Near-Fields from $$\mathfrak {p}$$-Adic Division Algebras

The aim of this work is the construction of a new class of disconnected locally compact near-fields. They are all Dickson near-fields and derived from finite-dimensional division algebras over local fields by means of strong couplings with a finite Abelian Dickson group consisting of inner automorphisms.

A connection between the number of subgroups and the order of a finite group

For a finite group [Formula: see text], we associate the quantity [Formula: see text], where [Formula: see text] is the subgroup lattice of [Formula: see text]. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of [Formula: see text] on the class of [Formula: see text]-groups of order [Formula: see text], where [Formula: see text] is an integer. We show that the set containing the quantities [Formula: see text], where [Formula: see text] is a finite (abelian) group, is dense in [Formula: see text] Finally, we consider [Formula: see text] to be a function on [Formula: see text] and we indicate some of its properties, the main result being the classification of finite abelian [Formula: see text]-groups [Formula: see text] satisfying [Formula: see text]

Inverse problems associated with subsequence sums in Cp ⊕ Cp

Let G be a finite abelian group and S be a sequence with elements of G. We say that S is a regular sequence over G if ∣SH∣ ⩽ ∣H∣ − 1 holds for every proper subgroup H of G, where SH denotes the subsequence of S consisting of all terms of S contained in H. We say that S is a zero-sum free sequence over G if 0 ∉ Ω(S), where Ω(S) ⊂ G denotes the set of group elements which can be expressed as a sum of a nonempty subsequence of S. In this paper, we study the inverse problems associated with Ω(S) when S is a regular sequence or a zero-sum free sequence over G = Cp ⊕ Cp, where p is a prime.

Group connectivity under 3‐edge‐connectivity

AbstractLet be two distinct finite Abelian groups with . A fundamental theorem of Tutte shows that a graph admits a nowhere‐zero ‐flow if and only if it admits a nowhere‐zero ‐flow. Jaeger et al in 1992 introduced group connectivity as an extension of flow theory, and they asked whether such a relation holds for group connectivity analogy. It was negatively answered by Hušek et al in 2017 for graphs with edge‐connectivity 2 for the groups and . In this paper, we extend their results to 3‐edge‐connected graphs (including both cubic and general graphs), which answers open problems proposed by Hušek et al and Lai et al. Combining some previous results, this characterizes all the equivalence of group connectivity under 3‐edge‐connectivity, showing that every 3‐edge‐connected ‐connected graph is ‐connected if and only if .

Remark on subgroup intersection graph of finite abelian groups

Abstract Let G be a finite group. The subgroup intersection graph Γ ( G ) \text{Γ}(G) of G is a graph whose vertices are non-identity elements of G and two distinct vertices x and y are adjacent if and only if | 〈 x 〉 ∩ 〈 y 〉 | > 1 |\langle x\rangle \cap \langle y\rangle |\gt 1 , where 〈 x 〉 \langle x\rangle is the cyclic subgroup of G generated by x. In this paper, we show that two finite abelian groups are isomorphic if and only if their subgroup intersection graphs are isomorphic.

The Number of Subgroup Chains of Finite Nilpotent Groups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

Self-dual codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ and applications

Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual linear codes over the ring~$\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ with $u^3=0$ have been established. In this paper, Hermitian self-dual linear codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ are studied for all square prime powers~$q$. Complete characterization and enumeration of such codes are given. Subsequently, algebraic characterization of $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ is studied, where $H\leq G$ are finite abelian groups and $\mathbb{F}_q[H]$ is a principal ideal group algebra. General characterization and enumeration of $H$-quasi-abelian codes and self-dual $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ are given. For the special case where the field characteristic is $3$, an explicit formula for the number of self-dual $A\times \mathbb{Z}_3$-quasi-abelian codes in $\mathbb{F}_{3^m}[A\times \mathbb{Z}_3\times B]$ is determined for all finite abelian groups $A$ and $B$ such that $3\nmid |A|$ as well as their construction. Precisely, such codes can be represented in terms of linear codes and self-dual linear codes over $\mathbb{F}_{3^m}+u\mathbb{F}_{3^m}+u^2\mathbb{F}_{3^m}$. Some illustrative examples are provided as well.

Exponential lower bounds on the generalized Erdős–Ginzburg–Ziv constant

For a finite abelian group G, the generalized Erdős–Ginzburg–Ziv constant sk(G) is the smallest m such that a sequence of m elements in G always contains a k-element subsequence which sums to zero. If n=exp(G) is the exponent of G, the previously best known bounds for skn(Cnr) were linear in n and r when k≥2. Via a probabilistic argument, we produce the exponential lower bound s2n(Cnr)>n2[1.25+o(1)]rfor n>0. For the general case, we show skn(Cnr)>kn4(1+1ek+1+o(1))r.

Finite Abelian Groups with the Strong Endomorphism Kernel Property

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.

Improved divisor arithmetic on generic hyperelliptic curves

The divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.

Conference matrices from Legendre C-pairs

Introduction: There are just a few known methods for the construction of symmetric C-matrices, due to the lack of a universal structure for them. This obstruction is fundamental, in addition, the structure of C-matrices with a double border is incompletely described in literature, which makes its study especially relevant. The purpose: To describe the two-border two-circulant construction in detail with the proposal of the concept of C-pairs Legendre. Results: The paper deals with C-matrices of order n=2v+2 with two borders and extends the so called generalized Legendre pairs, v odd, to a wider class of Legendre C-pairs with even and odd v, defined on a finite abelian group G of order v. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1 except that a(e)=0, where e is the identity element of G and Z is the ring of integers. To characterize the Legendre C-pairs we use the subsets X={xÎG: a(x)=–1} and Y={xÎG: b(x)=–1} of G. We show that a(x−1)=(−1)v a(x) for all x. For odd v we show that X and Y form a difference family, which is not true for even v. These difference families are precisely the so called Szekeres difference sets, used originally for the construction of skew-Hadamard matrices. We introduce the subclass of the special Legendre C-pairs and prove that they exist whenever 2v+1 is a prime power. In the last two sections of the paper we list examples of special cyclic Legendre C-pairs for lengths v<70. Practical relevance: C-matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of conference matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms.

Absolute Maximum Nonlinear Functions on Finite Nonabelian Groups

Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) between finite fields have been studied in numerous papers. These concepts have been generalized first to finite abelian groups and then to finite nonabelian groups. In 2011, Poinsot and Pott introduced a new class of highly nonlinear functions between finite nonabelian groups, which are closely related to maximum nonlinear functions. They found such a function by a computer search, and proposed to find more such functions by theoretical constructions. Since then there is no progress in this topic. In this paper we continue the research in the direction proposed by Poinsot and Pott. Our purpose is to study the properties and constructions of this new class of highly nonlinear functions, called the absolute maximum nonlinear functions in this paper. In particular, for arbitrary finite (abelian or nonabelian) groups $K$ and $N$ with an absolute maximum nonlinear function $f: K \to N$ , we will prove that $N$ has to be abelian, and $f$ cannot be perfect nonlinear if $K$ is not abelian. Then we investigate the numerical constraints on the groups that have absolute maximum nonlinear functions. These constraints will eliminate the existence of absolute maximum nonlinear functions for many finite nonabelian groups. More importantly, using these constraints we will develop a method to construct absolute maximum nonlinear functions for some groups. In particular, we will determine all absolute maximum nonlinear functions on $A_4$ (the alternating group of degree 4). Furthermore, a general method to construct new absolute maximum nonlinear functions from the old ones will be given.

A method to determine algebraically integral Cayley digraphs on finite abelian group

A method to determine algebraically integral Cayley digraphs on finite abelian group

On the lattice of subquandles of a Takasaki quandle

Let A be an abelian group. If for any we define , then is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by . Indeed, we characterize the subquandles of T(A) and show that is the lattice of cosets of A if and only if A is an odd abelian group or its 2-Sylow subgroup is a cyclic group. Further, we determine the homotopy type of the lattice of subquandles of T(A) which is the wedge of r spheres of dimension d, where r is the number of prime numbers dividing and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that is graded if and only if A is an odd abelian group or its 2-Sylow subgroup is cyclic or its 2-Sylow subgroup is elementary abelian. Moreover, we show that the lattice of subquandles of a finite Takasaki quandle is determined by a unique Takasaki quandle. In other words, we show that the lattices of subquandles of two finite Takasaki quandles T(A) and T(B) are isomorphic if and only if A and B are isomorphic. However, the lattice of subquandles of a quandle which is not a Takasaki quandle can be isomorphic to the lattice of subquandles of a Takasaki quandle. We show this by providing a quandle Q of order 8 which is not a Takasaki quandle, but is isomorphic to

On strongly sequenceable abelian groups

A group is strongly sequenceable if every connected Cayley digraph on the group admits an orthogonal directed cycle or an orthogonal directed path. This paper deals with the problem of whether finite abelian groups are strongly sequenceable. A method based on posets is used to show that if the connection set for a Cayley digraph on an abelian group has cardinality at most nine, then the digraph admits either an orthogonal directed path or an orthogonal directed cycle.

2-categories of symmetric bimodules and their 2-representations

In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting $2$-category is fiat (in the sense of \cite{MM1}) and classify simple transitive $2$-representations of this $2$-category (under some mild technical assumption). We also study several classes of examples in detail.

Separability of Schur Rings Over Abelian Groups of Odd Order

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. A finite group $G$ is said to be separable with respect to $\mathcal{K}$ if every $S$-ring over $G$ is separable with respect to $\mathcal{K}$. We prove that every abelian group $G$ of order $9p$, where $p$ is a prime, is separable with respect to the class of all finite abelian groups. Modulo previously obtained results, this completes a classification of noncyclic abelian groups of odd order that are separable with respect to the class of all finite abelian groups. Also this implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.