Abelian Groups In Terms Research Articles

Additive Complementary Dual Codes From Group Characters

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \leq 10$ </tex-math></inline-formula> . The obtained codes are either optimal or near optimal according to Bierbrauer <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> +. (2009).

Self-dual additive codes

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group $${\mathbb {Z}}_{p^e}$$ . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with $$p \equiv 1 \pmod {4}$$ and e is odd with $$e>1$$ ; and all lengths that are $$0 \pmod {4}$$ when p is an odd prime with $$p \equiv 3 \pmod {4}$$ and e is odd with $$e>1.$$

Approximations for Theories of Abelian Groups

Approximations of syntactic and semantic objects play an important role in various fields of mathematics. They can create theories and structures in one given class by means of others, usually simpler. For instance, in certain situations, infinite objects can be approximated by finite or strongly minimal ones. Thus, complicated objects can be collected using simplified ones. Among these objects, Abelian groups, their first order theories, connections and dynamics are of interests. Theories of Abelian groups are characterized by Szmielew invariants leading to the study and descriptions of approximations in terms of these invariants. In the paper we apply a general approach for approximating theories to the class of theories of Abelian groups which characterizes the approximability of a theory of Abelian groups by a given family of theories of Abelian groups in terms of Szmielew invariants and their limits. We describe some forms of approximations for theories of Abelian groups. In particular, approximations of theories of Abelian groups by theories of finite ones are characterized. In addition, we describe approximations by quasi-cyclic and torsion-free Abelian groups and their combinations with respect to given families of prime numbers. Approximations and closures of families of theories with respect to standard Abelian groups for various sets of prime numbers are also described.

On 5–manifolds with free fundamental group and simple boundary links in S5

We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply this to the classification of simple boundary links of $3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed $5$-manifolds with free fundamental group and trivial second homology group.

Abelian groups whose nonzero endomorphisms have nonessential kernels

In this paper, we describe completely the [Formula: see text]-singular subgroup of an abelian group and a [Formula: see text]-nonsingular abelian group in terms of the basic subgroups of its [Formula: see text]-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a [Formula: see text]-nonsingular abelian group are [Formula: see text]-nonsingular and give a condition under which a pure extension of a [Formula: see text]-nonsingular abelian group by a [Formula: see text]-nonsingular group is [Formula: see text]-nonsingular.

Improved Bounds on Sidon Sets via Lattice Packings of Simplices

A $ B_h $ set (or Sidon set of order $ h $) in an Abelian group $ G $ is any subset $ \{b_0, b_1, \ldots,b_{n}\} $ of $ G $ with the property that all the sums $ b_{i_1} + \cdots + b_{i_h} $ are different up to the order of the summands. Let $ \phi(h,n) $ denote the order of the smallest Abelian group containing a $ B_h $ set of cardinality $ n + 1 $. It is shown that \[ \lim_{h \to \infty} \frac{ \phi(h,n) }{ h^n } = \frac{1}{n! \delta_L(\triangle^n)} , \] where $ \delta_L(\triangle^n) $ is the lattice packing density of an $ n $-simplex in Euclidean space. This determines the asymptotics exactly in cases where this density is known ($ n \leq 3 $) and gives improved bounds on $ \phi(h,n) $ in the remaining cases. The corresponding geometric characterization of bases of order $ h $ in finite Abelian groups in terms of lattice coverings by simplices is also given.

Cotorsion-free groups from a topological viewpoint

We present a characterization of cotorsion-free abelian groups in terms of homomorphisms from fundamental groups of Peano continua, which aligns naturally with the generalization of slenderness to non-abelian groups. In the process, we calculate the first homology group of the Griffiths twin cone.

On the powers of maximal ideals in the measure algebra

In this paper, we describe the powers of maximal ideals in the measure algebra of some locally compact Abelian groups in terms of the derivatives of the Fourier–Laplace transform of compactly supported measures. We show that if the locally compact Abelian group has sufficiently many real characters, then all derivatives of the Fourier–Laplace transform of a measure at some point of its spectrum completely characterize the measure. We also show that the derivatives of the Fourier–Laplace transform of a measure can be used to describe the powers of the maximal ideals corresponding to the points of the spectrum of the measure on discrete Abelian groups with finite torsion-free rank.

A Note on OI Torsion Abelian Groups

Let $R$ be a commutative ring with unity and $M_R$ be a nonzero unital right $R$-module. We say that $M$ is an OI $R$-module if for each $x \in R$, $Mx = M$ implies $x$ is invertible in $R$. We give a characterization of OI torsion abelian groups in terms of their direct summands.

Measurability in modules

In this paper we prove that in modules, MS-measurability (in the sense of Macpherson---Steinhorn) depends on being able to define a measure function on the p.p. definable subgroups. We give a classification of abelian groups in terms of measurability. Finally we discuss the relation with $${\mathbb{Q}[t]}$$ Q [ t ] -valued measures.

Polynomial Functors from Algebras Over a Set-Operad and Nonlinear Mackey Functors

In this paper, we give a description of polynomial functors from ( finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on ( finitely generated free) monoids and those on ( finitely generated free) groups. Polynomial functors from ( finitely generated free) P-algebras to abelian groups and from ( finitely generated free) groups to abelian groups are described explicitely by their cross-e ffects and maps relating them which satisfy a list of relations.

Critical Groups of Simplicial Complexes

We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups.

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C ( G ) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C ( G ) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C ( G ) ∖ { G } is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T , S we define the equalizer E ( T , S ) , which permits to describe relevant classes of abelian groups in terms of functorial topologies.

Critical Groups of Simplicial Complexes

We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups. Nous généralisons la théorie des groupes critiques des graphes aux complexes simpliciaux. Plus précisément, pour un complexe simplicial, nous définissons une famille de groupes abéliens en termes d'opérateurs de Laplace combinatoires, qui généralise la construction du groupe critique d'un graphe. Nous montrons comment réaliser ces groupes critiques explicitement comme conoyaux des opérateurs de Laplace réduits combinatoires, et montrons qu'ils sont finis. Leurs ordres sont obtenus en comptant (avec des poids) des arbres simpliciaux couvrants. Nous décrivons comment les groupes critiques d'un complexe représentent le flux le long de ses faces, et esquissons une autre interprétation potentielle comme analogues des groupes de Chow.

On the Jacobson radical of the endomorphism ring of a homogeneous separable group

Pierce [1] posed the problem of describing the elements of the Jacobson radical of the endomorphism ring of a primary Abelian group in terms of actions of these elements on the group. The problem of describing the Jacobson radical of the endomorphism ring of a homogeneous separable group was posed in this sense in [2, Problem 18]. In the present note, we study the Jacobson radical of the endomorphism ring of a homogeneous separable torsion-free group, and the above problem is solved (see Theorem 4 and Theorem 8). All groups under consideration are Abelian torsion-free groups. Let G be a group; denote by E(G) its endomorphism ring. Denote by H(G) the set of all endomorphisms α ∈ E(G) such that αx =0 for any x ∈ D(G) ,w hereD(G) is the divisible part of the group G and, for any x ∈ G \ D(G), the inequality hp(x) h p(x) ,w herehp(x) stands for the p-height of the element x. Denote by F (G) the set of all endomorphisms of G whose image has finite rank. The Jacobson radical of the endomorphism ring of G is denoted by J(E(G)). In what follows, we consider torsion-free Abelian groups only. For the notation and terms which are not defined in this note; see [2] and [3]. As is known [2, Sec. 22], if G is a homogeneous completely decomposable group, then J(E(G)) = H(G) ∩ F (G). For a homogeneous separable group, only the following inclusion relations are known: H(G) ∩ F (G) ⊆ J(E(G)) ⊆ H(G).

John-type theorems for generalized arithmetic progressions and iterated sumsets

A classical theorem of Fritz John allows one to describe a convex body, up to constants, as an ellipsoid. In this article we establish similar descriptions for generalized (i.e. multidimensional) arithmetic progressions in terms of proper (i.e. collision-free) generalized arithmetic progressions, in both torsion-free and torsion settings. We also obtain a similar characterization of iterated sumsets in arbitrary abelian groups in terms of progressions, thus strengthening and extending recent results of Szemerédi and Vu.

Expansive algebraic actions of discrete residually finite amenable groups and their entropy

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma$ by automorphisms of compact abelian groups in terms of Fuglede?Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma$-action by means of a ?fundamental homoclinic point? and the description of entropy in terms of the renormalized logarithmic growth rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n,n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma$ with trivial intersection.

On the set of integral solutions of the Pell equation in number fields

We investigate the set of integral solutions, over a given number field, of the equation X2 - dY2= 1, where d denotes some non-zero integer of this field. We define an operation on this set such that it is an abelian group and determine the structure of this abelian group in terms of the number of complex and real embeddings of the number field, and the number of positive embeddings of d.

Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups

We use additive notation, +, 0, ',-, for Abelian groups, and multiplicative notation, , e, l, /, for ordinary groups. Throughout this note, and / are binary operations rather than abbreviations for, e.g., x + y' and x Xy-1 One might think it trivial, given (2), to obtain a single axiom in terms of product and inverse, by simply rewriting a/,(3 to a * l Doing so gives a single axiom, but then * is not product, and -1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and the identity element [8]. Single equational axioms in terms of product and inverse have been reported by Neumann [5] and others [3, 2]. In this note we consider single amplicative axioms, that is, axioms of the form ct = ,X3 y = b. For Abelian groups, an axiom of this type with five variables was

Multidimensional Cooley–Tukey Algorithms Revisited

The representation theory of Abelian groups is used to obtain an algebraic divide-and-conquer algorithm for computing the finite Fourier transform. The algorithm computes the Fourier transform of a finite Abelian group in terms of the Fourier transforms of an arbitrary subgroup and its quotient. From this algebraic algorithm a procedure is derived for obtaining concrete factorizations of the Fourier transform matrix in terms of smaller Fourier transform matrices, diagonal multiplications, and permutations. For cyclic groups this gives as special cases the Cooley–Tukey and Good–Thomas algorithms. For groups with several generators, the procedure gives a variety of multidimensional Cooley–Tukey type algorithms. This method of designing multidimensional fast Fourier transform algorithms gives different data flow patterns from the standard “row–column” approaches. We present some experimental evidence that suggests that in hierarchical memory environments these data flows are more efficient.