Finite Abelian Research Articles

Comments on “Extremal Cayley Digraphs of Finite Abelian Groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135

We comment on the paper “Extremal Cayley digraphs of finite Abelian groups” [Intercon. Networks 12 (2011), no. 1-2, 125–135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.

Nil-clean matrix rings

We characterize the nil-clean matrix rings over fields. As a by product, we obtain a complete characterization of the finite rank Abelian groups with nil-clean endomorphism ring and the Abelian groups with strongly nil-clean endomorphism ring, respectively.

Some remarks on barycentric-sum problems over cyclic groups

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer ℓ such that each subset A of G with at least ℓ elements contains a subset with k elements {g1,…,gk} satisfying g1+⋯+gk=kgj for some 1≤j≤k.

On the Fast Computation of the Weight Enumerator Polynomial and the $t$ Value of Digital Nets over Finite Abelian Groups

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams-type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$-dimensional unit cube $[0,1)^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further.

Implement an effective technique for numerical solution of problems of electronic optics

Some aspects of the numerical solution of boundary value problems of potential theory with finite order Abelian groups of symmetry is considered. An effective method based on the method of integral equations in conjunction with the apparatus of group theory

A New Construction of Jacket Matrix Based on Finite Abelian Group

A New Construction of Jacket Matrix Based on Finite Abelian Group

Identities of PI-Algebras Graded by a Finite Abelian Group

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ℤ2)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.

On the Number of Subsequences with a Given Sum in a Finite Abelian Group

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S)\ge2^{|S|-D(G)+1}$, where $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.

The universality theorem for Hecke L-functions

We extend the universality theorem for Hecke L-functions attached to ray class characters from the previously known strip \({ \max \{\frac{1}{2}, 1-\frac{1}{d}\} < {\rm Re}\,s < 1}\) for \({d=\left[K:\mathbb{Q}\right]}\) to the maximal strip \({\frac{1}{2} < {\rm Re}\,s < 1}\) under an assumption of a weak version of the density hypothesis. As a corollary, we give a new proof of the universality theorem for the Dedekind zeta function ζK(s) in the case of \({K/\mathbb{Q}}\) finite abelian.

Monochrome Symmetric Subsets in Colorings of Finite Abelian Groups

A subset S of a group G is symmetric if there is an element g є G such that gS-1g = S. We study some Ramsey type functions for symmetric subsets in finite Abelian groups.

Characteristic Subgroups of Finite Abelian Groups

We consider the following question: When do two finite abelian groups have isomorphic lattices of characteristic subgroups? An explicit description of the characteristic subgroups of such groups enables us to give a complete answer to this question, in the case where at least one of the groups has odd order. An “exceptional” isomorphism, which occurs between the lattice of characteristic subgroups of Z p × Z p 2 × Z p 4 and Z p 2 × Z p 5 , for any prime p, is noteworthy.

A course on abstract algebra

Algebraic Structure of Numbers Sets and Functions Basic Group Theory Elementary Group Theory Group Homomorphisms Direct Products of Finite Abelian Groups Sylow's Theorems and Applications Symmetric Groups Basic Ring Theory Special Properties of Rings Field Theory Basic Galois Theory.

Duality of the categories of torsion-free Abelian groups of finite rank and quotient divisible Abelian groups

A new approach to the description of quotient divisible Abelian groups is proposed. This approach allowed us to give an explicit and natural proof of the duality of the categories of torsion-free Abelian groups of finite rank and quotient divisible Abelian groups with distinguished basic subgroups. Bibliography: 3 titles.

On the precise order of unit groups of burnside rings of some finite Abelian groups

On the precise order of unit groups of burnside rings of some finite Abelian groups

Group Algebras of Finite Abelian Groups and their Applications to Combinatorial Problems

Let G be an additive finite abelian group. In the last decades group algebras R[G] over suitable commutative rings R have turned out to be powerful tools for a growing variety of questions from combinatorics and number theory. Many of them can be reduced to the problem whether for some given sequence S = g1 · . . . · gl over G the elements f = (X1 − a1) · . . . · (Xl − al) 6= 0 ∈ R[G] for all a1, . . . , al ∈ R {0} . The present paper is devoted to this crucial problem. Before presenting our new results we recall the classical application of group algebras to the investigation of zero-sumfree sequences which is due to P. van Emde Boas, D. Kruyswijk and J.E. Olson (see [8], [9],[20]). Let d(G) denote the maximal length of a zero-sumfree sequence over G. Then d(G)+ 1 is the Davenport constant of G. For some commutative ring R, let d(G,R) denote the largest integer l ∈ N having the following property: there is some sequence S over G of length |S| = l such that (X1 − a1) · . . . · (Xl − al) 6= 0 ∈ R[G] for all a1, . . . , al ∈ R {0} . If S is zero-sumfree, R an integral domain, a1, . . . , al ∈ R {0} and

The Capacity of Finite Abelian Group Codes Over Symmetric Memoryless Channels

The capacity of finite Abelian group codes over symmetric memoryless channels is determined. For certain important examples, such as m -PSK constellations over additive white Gaussian noise (AWGN) channels, with m a prime power, it is shown that this capacity coincides with the Shannon capacity; i.e., there is no loss in capacity using group codes. (This had previously been known for binary-linear codes used over binary-input output-symmetric memoryless channels.) On the other hand, a counterexample involving a three-dimensional geometrically uniform constellation is presented in which the use of Abelian group codes leads to a loss in capacity. The error exponent of the average group code is determined, and it is shown to be bounded away from the random-coding error exponent, at low rates, for finite Abelian groups not admitting Galois field structure.

Complexity andT‐invariant of Abelian and Milnor groups, and complexity of 3‐manifolds

We investigate the notion of complexity for finitely presented groups and the related notion of complexity for three-dimensional manifolds. We give two-sided estimates on the complexity of all the Milnor groups (the finite groups with free action on S3), as well as for all finite Abelian groups. The ideas developed in the process also allow to construct two-sided bounds for the values of the so-called T -invariant (introduced by Delzant) for the above groups, and to estimate from below the value of T -invariant for an arbitrary finitely presented group. Using the results of this paper and of previous ones, we then describe an infinite collection of Seifert threemanifolds for which we can asymptotically determine the complexity in an exact fashion up to linear functions. We also provide similar estimates for the complexity of several infinite families of Milnor groups. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Unextendible Sequences in Finite Abelian Groups

Let $G=C_{n_1}\oplus \ldots \oplus C_{n_r}$ be a finite abelian group with $r=1$ or $1 &lt; n_1|\ldots|n_r$, and let $S=(a_1,\ldots,a_t)$ be a sequence of elements in $G$. We say $S$ is an unextendible sequence if $S$ is a zero-sum free sequence and for any element $g\in G$, the sequence $Sg$ is not zero-sum free any longer. Let $L(G)=\lceil \log_2{n_1}\rceil+\ldots+\lceil \log_2{n_r}\rceil$ and $d^*(G)=\sum_{i=1}^r(n_i-1)$, in this paper we prove, among other results, that the minimal length of an unextendible sequence in $G$ is not bigger than $L(G)$, and for any integer $k$, where $L(G)\leq k \leq d^*(G)$, there exists at least one unextendible sequence of length $k$.

Automorphisms of Finite Abelian Groups

(2007). Automorphisms of Finite Abelian Groups. The American Mathematical Monthly: Vol. 114, No. 10, pp. 917-923.

On the Number of Subsequences with Given Sum of Sequences over Finite Abelian $p$-Groups

Let G be an additive finite abelian p-group. For a given (long) sequence S in G and some element g ∈ G, we investigate the number of subsequences of S which have sum g. This refines some classical results of J.E. Olson and recent results of I. Koutis.