Abelian Group Research Articles

Simple-separable modules

A module $M$ over a ring is called simple-separable if every simple submodule of $M$ is contained in a finitely generated direct summand of $M$. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module $M$ over a right noetherian ring is simple-separable if and only if $M=M_1 \oplus M_2$ such that $M_1$ is separable and $M_2$ has zero socle. The structure of simple-separable abelian groups is completely described.

Restricted-finite groups with some applications in group rings

We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to $ \mathbb{Z}\times K$ or $\mathbb{Z}_{p^\infty}\times K$, where $K$ is a finite group and $p$ is a prime number. We also prove that a group $G$ is infinitely generated restricted-finite if and only if $G = AT$ where $A$ and $T$ are subgroups of $G$ such that $A$ is normal quasi-cyclic and $T$ is finite. As an application of our results, we show that if $G$ is not torsion with finite $G'$ and the group-ring $RG$ has restricted minimum condition, then $R$ is a semisimple ring and $G\cong T\rtimes\mathbb{Z} $, where $T$ is finite whose order is unit in $R$. The converse is also true with certain conditions including $G = T\times \mathbb{Z} $.

Pure point diffraction and entropy beyond the Euclidean space

For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of σ-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. For this we provide a suitable version of the variational principle. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set cannot be computed along a limit and even be infinite in this setting.

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic representation theory of finite groups indicates that a function f on a finite abelian group G can be written as a linear combination of characters of irreducible representations of G by f(x)=∑χ∈Gˆfˆ(χ)χ(x), where Gˆ is the dual group of G consisting of all characters of G and fˆ(χ) is the Fourier coefficient of f at χ∈Gˆ. In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of f on a finite abelian group G with complexity that is quasi-linear in the order of G and polynomial in the FSOS sparsity of f. Moreover, for a nonnegative function f on a finite abelian group G and a subset S⊆Gˆ, we give a lower bound of the constant M such that f+M admits an FSOS supported on S. We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to 107. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem by sparse FSOS.

Owings-like theorems for infinitely many colours or finite monochromatic sets

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers κ,θ, every colouring c:G⟶θ yields a subset X⊆G with |X|=κ such that X+X is monochromatic. (Owings's problem asks this for G=Z, θ=2 and κ=ℵ0; this is known to be false for the same G and κ but θ=3.) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.

Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g\geq 3$.

Association Schemes for Some Finite Group Rings II

Association schemes have been used in coding theory and other combinatorial problems. In this paper, we construct association schemes for the abelian groups Z_2^r,〖 Z〗_(n_1 )×Z_(n_2 )×⋯×Z_(n_r ), set of n ×n matrices over Z_m and for the general linear group of order 2 over Z_2,Z_4, and Z_6. We also obtain association schemes for symmetric groups and alternating groups of degree 4 and 5 using canonical forms.

Ax's lemma and Ax's theorem in the Aichinger-Moosbauer calculus

In 1964, J. Ax gave a ten line proof of the Chevalley–Warning Theorem. The novelty was an innocuous result about summing a polynomial f∈Fq[t1,…,tN] over all x∈FqN. Recent work of Aichinger–Moosbauer provides a context in which we can seek to generalize Ax's Lemma to maps f:A→B between any two finite commutative groups, and indeed Aichinger–Moosbauer proved such a result when the target group B has prime exponent. In this paper we define a summation invariantσ(A,B) that fits naturally into the Aichinger–Moosbauer calculus. We give upper and lower bounds on σ(A,B) and its exact value in many (but not all) cases. We also give Diophantine applications, including a qualitative Ax–Katz Theorem for polynomials over any finite rng. The bounds that we get turn out to be closely related to Ax's part of the Ax–Katz Theorem.

On Multilinear Inequalities of Ho ̈lder-Brascamp-Lieb Type for Torsion-Free Discrete Abelian Groups

Holder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of L^p norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequalities for torsion-free discrete Abelian groups in three respects.The optimal constant in any such inequality is shown to equal 1 whenever it is finite.An algorithm that computes the admissible polyhedron of exponents is developed. It is shown that nonetheless, existence of an algorithm that computes the full list of inequalitiesin the Bennett-Carbery-Christ-Tao description of the admissible polyhedron for all data,is equivalent to an affirmative solution of Hilbert's Tenth Problem over the rationals.That problem remains open.

مونويدات جزئية من الزمر الباراتبولوجية الآبيلية

One of the important subclasses of abelian paratopological groups is called the free abelian paratopological group on a topological space. It was introduced in 2003 by Remaguara, Sanchis, and Tkachenko. In this paper, we introduce a single submonoid of the free abelian paratopological group on Alexandroff space, then we prove that this submonoid is a base at the identity element of the free topology of the abelian group. The main result of this paper is to give applications of this submonoid such as studying separation axioms, compactness, and other properties of the free topology on the abelian group. Keywords: free group, Abelian paratopological group, free abelian paratopological group, Alexandroff space, neighborhood base at an identity.

Ergodic theorem for nonstationary random walks on compact abelian groups

We consider a nonstationary random walk on a compact metrizable abelian group. Under a classical strict aperiodicity assumption we establish a weak-* convergence to the Haar measure, Ergodic Theorem and Large Deviation Type Estimate.

Groups Generated by Pattern Avoiding Permutations

We study groups generated by sets of pattern avoiding permutations. In the first part of the paper, we prove some general results concerning the structure of such groups. In particular, we consider the sequence ( G n ) n ≥ 0 , where G n is the group generated by a subset of the symmetric group S n consisting of permutations that avoid a given set of patterns. We analyze under which conditions the sequence ( G n ) n ≥ 0 is eventually constant. Moreover, we find a set of patterns such that ( G n ) n ≥ 0 is eventually equal to an assigned symmetric group. Furthermore, we show that any non-trivial simple group cannot be obtained in this way and describe all the non-trivial abelian groups that arise in this way. In the second part of the paper, we carry out a case-by-case analysis of groups generated by permutations avoiding a few short patterns.

Moduli spaces of ℤ/kℤ-constellations over 𝔸2

Let [Formula: see text] be a representation of a finite abelian group and let [Formula: see text] be the space of generic stability conditions on the set of [Formula: see text]-constellations. We provide a combinatorial description of all the chambers [Formula: see text] and prove that there are [Formula: see text] of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know all toric [Formula: see text]-constellations, it is enough to build all simple chambers. We also prove that there are [Formula: see text] simple chambers. Finally, we provide an explicit formula for the tautological bundles [Formula: see text] over the moduli spaces [Formula: see text] for all chambers [Formula: see text] which only depends upon the chamber stair which is a combinatorial object attached to the chamber [Formula: see text].

Convex Characteristics of Quaternionic Positive Definite Functions on Abelian Groups

Convex Characteristics of Quaternionic Positive Definite Functions on Abelian Groups

Perfect codes in 2-valent Cayley digraphs on abelian groups

For a digraph Γ, a subset C of V(Γ) is a perfect code if C is a dominating set such that every vertex of Γ is dominated by exactly one vertex in C. In this paper, we classify strongly connected 2-valent Cayley digraphs on abelian groups admitting a perfect code, and determine completely all perfect codes of such digraphs.

Descent of splendid Rickard equivalences in alternating groups

We show that any block of an alternating group with an abelian defect group over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent. This provides new evidence for a refined version of Broué's abelian defect group conjecture proposed by Kessar and Linckelmann.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

A Gabor orthonormal basis, on a locally compact Abelian (LCA) group A, is an orthonormal basis of L2(A) that consists of time-frequency shifts of some template f∈L2(A). It is well known that, on Rd, the elements of such a basis cannot have a good time-frequency localization. The picture is drastically different on LCA groups containing a compact open subgroup, where one can easily construct examples of Gabor orthonormal bases with f maximally localized, in the sense that the ambiguity function of f (i.e., the correlation of f with its time-frequency shifts) has support of minimum measure, compatibly with the uncertainty principle. In this note we find all the Gabor orthonormal bases with this extremal property. To this end, we identify all the functions in L2(A) that are maximally localized in the time-frequency space in the above sense – an issue that is open even for finite Abelian groups. As a byproduct, on every LCA group containing a compact open subgroup we exhibit the complete family of optimizers for Lieb's uncertainty inequality, and we also show previously unknown optimizers on a general LCA group.

Super Bassian and nearly generalized Bassian Abelian groups

In connection to two recent publications of ours in Arch. Math. Basel (2021) and Acta Math. Hungar. (2022), respectively, and in regard to the results obtained in Arch. Math. Basel (2012), we have the motivation to study the near property of both Bassian and generalized Bassian groups. Concretely, we prove that if an arbitrary reduced group has the property that all of its proper subgroups are generalized Bassian, then it is also generalized Bassian itself, that is, if [Formula: see text] is an arbitrary nearly generalized Bassian group, then either [Formula: see text] is a quasi-cyclic group or [Formula: see text] is generalized Bassian of finite torsion-free rank. Moreover, we show that there is a complete description of the so-called super Bassian groups that are those groups whose epimorphic images are all Basian, and give their full characterization. Also, we establish that the hereditary property of Bassian groups gives nothing new as it coincides with the ordinary Bassian property.

A note on matrix-valued orthonormal bases over LCA groups

It is well known that orthonormal bases for a separable Hilbert space [Formula: see text] are precisely collections of the form [Formula: see text], where [Formula: see text] is a linear unitary operator acting on [Formula: see text] and [Formula: see text] is a given orthonormal basis for [Formula: see text]. We show that this is not true for the matrix-valued signal space [Formula: see text], [Formula: see text] is a locally compact abelian group which is [Formula: see text]-compact and metrizable, and [Formula: see text] and [Formula: see text] are positive integers. This problem is related to the adjointability of bounded linear operators on [Formula: see text]. We show that any orthonormal basis of the space [Formula: see text] is precisely of the form [Formula: see text], where [Formula: see text] is a linear unitary operator acting on [Formula: see text] which is adjointable with respect to the matrix-valued inner product and [Formula: see text] is a matrix-valued orthonormal basis for [Formula: see text].

Non-Virtually Abelian Discontinuous Group Actions vs. Proper SL(2,ℝ) -Actions on Homogeneous Spaces

We develop algorithms and computer programs which verify criteria of properness of discrete group actions on semisimple homogeneous spaces. We apply these algorithms to find new examples of non-virtually abelian discontinuous group actions on homogeneous spaces which do not admit proper SL ( 2 , R ) -actions.