On the order of additive bases in finite abelian groups

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

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

Dynamical behavior of additive cellular automata over finite abelian groups

Sum-free subsets of finite abelian groups of type III

This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an additional application of this characterization, we show that a variety with abelian quotient singularities may fail to be a quotient of a smooth variety by a finite abelian group. Concretely, we show that $\mathbb{P}^2/A_5$ is not expressible as a quotient of a smooth variety by a finite abelian group.

Let G≅C n 1 ⊕...⊕C n r be a finite and nontrivial abelian group with n 1 |n 2 |...|n r . A conjecture of Hamidoune says that if W=w 1 ·...·w n is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S|≥|W|+|G|-1≥|G|+1, the maximum multiplicity of S at most |W|, and σ(W)≡0mod|G|, then there exists a nontrivial subgroup H such that every element g∈H can be represented as a weighted subsequence sum of the form g=∑ n i=1w i s i , with s 1 ·...·s n a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W|≥1 2|G|.A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S|≥|G|+𝔻(G)-1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g+H such that all but at most |G/H|-2 terms of S are from g+H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S|≥|G|+𝔻(G)-1 can be relaxed to |S|≥|G|+d * (G), where d * (G)=∑ r i=1(n i -1). We also use this method to derive a variation on Hamidoune’s conjecture valid when at least d * (G) of the w i are relatively prime to |G|.

A famous conjecture of Minkowski, concerning the columnation of space-filling lattices, was first proved by Hajos in 1941 by translating the problem into one involving finite abelian groups. The problem solved by Hajos was one concerning a special type of factorisation of finite abelian groups. In the general problem considered in the thesis no restriction is placed on the nature of the factors. It was originally conjectured by Hajos that in any factorisation; one of the factors must possess a non-trivial subgroup as a factor, However, Hajos himself soon found that not all finite abelian groups possess this property. Those which do were called "good" and those which do not were called "bad" Further contributions to determining those groups which are good and those which are had were made by Redei and de Bruijn. But for groups of many types the problem was left undecided. In this thesis the problem is solved completely for finite abelian groups. A special case of this problem for cyclic groups was shown by de Bruijn to be equivalent to a conjecture of his concerning bases for the sets of integers. This conjecture and a generalisation of it are also shown to be true. It is shown first that a cyclotomic polynomial is irreducible over certain fields of reots of unity. This extension of the well-known result that a eyclotomic polynomial is irreducible over the rational field is basic to the following work and is used frecuently throughout the thesis. A theorem, similer to the theorems of de Bruijn, showing that certain types of groups are had is then proved, then, in the main part of the thesis all the groups not shown to be bad by this theorem or one of the theorems of de Bruijn are shown to be good. Hajos gave a method which, be claimed, would give all factorisations of a good group. However it is shown that a correction is needed in this method and the corrected method is then presented. The final section is concerned with the extension of the results to certain types of infinite abelian groups. Under the restriction that one of the factors shall have only a finite number of elements, similar results to these proved for finite groups are obtained for the generalisations of these groups to the infinite cases.

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p r , i.e., a finite homocyclic abelian group. Let Δ n (G) denote the n-th power of the augmentation ideal Δ(G) of the integral group ring ℤG. The paper gives an explicit structure of the consecutive quotient group Q n (G) = Δ n (G)/Δ n+1(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.

The theory of Gabor frames of functions defined on finite abelian groups was initially developed in order to better understand the properties of Gabor frames of functions defined over the reals. However, during the last twenty years the topic has acquired an interest of its own. One of the fundamental questions asked in this finite setting is the existence of full spark Gabor frames. The author proved the existence, as well as constructed such frames, when the underlying group is finite cyclic. In this paper, we resolve the non-cyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian non-cyclic groups. We also prove that all eigenvectors of certain unitary matrices in the Clifford group in odd dimensions generate spark deficient Gabor frames. Finally, similarities between the uncertainty principles concerning the finite dimensional Fourier transform and the short-time Fourier transform are discussed.

Let S be a fine and saturated (fs) log scheme, and let F be a group scheme over the underlying scheme of S which is étale locally representable by (1) a finite dimensional $\mathbb{Q}$ -vector space, or (2) a finite rank free abelian group, or (3) a finite abelian group. We give a full description of all the higher direct images of F from the Kummer log flat site to the classical flat site. In particular, we show that: in case (1) the higher direct images of F vanish; and in case (2) the first higher direct image of F vanishes and the nth ( $n\gt 1$ ) higher direct image of F is isomorphic to the $(n-1)$ -th higher direct image of $F\otimes_{{\mathbb Z}}{\mathbb Q}/{\mathbb Z}$ . In the end, we make some computations when the base is a standard henselian log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.

Keller's conjecture for certain finite groups

Let $G$ be a finite abelian group of order $n$ and $\mathcal M_G$ the Cayley table of $G$. Let $\mathcal P(G)$ be the number of formally different monomials occurring in $\mathsf {per}(\mathcal M_G)$, the permanent of $\mathcal M_G$. In this paper, for any finite abelian groups $G$ and $H$, we prove the following characterization $$\mathcal P(G)=\mathcal P(H)\ \Leftrightarrow\ G\cong H.$$ It follows that the group permanent determines the finite abelian group, which partially answers an open question of Donovan, Johnson and Wanless. In fact, $\mathcal P(G)$ is closely related to zero-sum sequences over finite abelian groups and we shall prove the above characterization by studying a reciprocity of zero-sum sequences over finite abelian groups. As an application of our method, we show that $\mathcal P(G)>\mathcal P(C_n)$ for any non-cyclic abelian group $G$ of order $n$ and thereby answer an open problem of Panyushev.

On the number of gradings on matrix algebras

Introduction Cast of characters Part I: 1. Congruences and the quotient ring of the integers mod n 1.2 The discrete Fourier transform on the finite circle 1.3 Graphs of Z/nZ, adjacency operators, eigenvalues 1.4 Four questions about Cayley graphs 1.5 Finite Euclidean graphs and three questions about their spectra 1.6 Random walks on Cayley graphs 1.7 Applications in geometry and analysis 1.8 The quadratic reciprocity law 1.9 The fast Fourier transform 1.10 The DFT on finite Abelian groups - finite tori 1.11 Error-correcting codes 1.12 The Poisson sum formula on a finite Abelian group 1.13 Some applications in chemistry and physics 1.14 The uncertainty principle Part II. Introduction 2.1 Fourier transform and representations of finite groups 2.2 Induced representations 2.3 The finite ax + b group 2.4 Heisenberg group 2.5 Finite symmetric spaces - finite upper half planes Hq 2.6 Special functions on Hq - K-Bessel and spherical 2.7 The general linear group GL(2, Fq) 2.8. Selberg's trace formula and isospectral non-isomorphic graphs 2.9 The trace formula on finite upper half planes 2.10 The trace formula for a tree and Ihara's zeta function.

It is proved that every finite lattice is isomorphic to an R-congruence lattice of a finite unar (finite Abelian group), as well as to a lattice of R-varieties for some locally finite finitely axiomatizable quasivariety of unars (Abelian groups) R.