On finite groups with cyclic Abelian subgroups

Solvable and minimal unsolvable finite groups with cyclic Abelian subgroups are constructively described. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 739–741, May, 1998.     

On the enumeration of finite Abelian and solvable groups

Let a(n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a(n). More generally, we get short interval version of results of Ivi? on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdös and Szekeres on the summatory function of a(n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups.     

An addition theorem for finite Abelian groups

If g₁, g₂, …, g_2n?1 is a sequence of 2n ? 1 elements in an Abelian group G of order n, it is known that there are n distinct indices i₁, i₂, …, i_n such that 0 = g_i1 + g_i2 + ? + g_in. In this paper a suitably general condition on the sequence is given which insures that every element g in G has a representation g = g_i1 + g_i2 + ? + g_in as the sum of n terms of the sequence.     

Differences of functions and groups generators for compact Abelian groups

LetG be a Hausdorff compact Abelian group andC be the component of the identity element ofG. We consider a special class, ?(G), of functions inL ² (G) whose Fourier series satisfy certain convergence conditions (stronger than absolute convergence). We show thatG/C is topologically generated by not more thann elements if and only if, for each functionf in ?(G), there area ₁,...,a _n inG and functionf ₁,...f _n in ?(G) such that $$f = \sum\limits_{j = 1}^n {(f_j - \delta _{aj} * f_j ),}$$ where * is convolution defined in the usual sense, and δ _a denotes the Dirac measure ataεG.     

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

We define the multiple zeta function of the free Abelian group Z^d as

ζ_Z^d(s₁,…,s_d)=∑_|Z^d:H|<∞α₁(H)^−s₁?α_d(H)^−s_d,     

Gabor analysis over finite Abelian groups

Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time–frequency plane, are the central topic of this paper. Our generic approach covers both multi-dimensional signals as well as non-separable lattices, and in fact the multi-window case as well. Our generic approach includes most of the fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler–Raz or Tolimieri–Orr, Bastiaans and Van-Leest and others. In our presentation the spreading representation of linear operators between finite-dimensional Hilbert space as well as a symplectic version of Poisson's summation formula over the finite time–frequency plane are essential ingredients. They bring us to the so-called Fundamental Identity of Gabor Analysis. In addition, we highlight projective representations of the time–frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting discussed in this paper these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames, independent of the structure theory theorem for finite Abelian groups.     

A generalization of the Legendre symbol for finite Abelian groups

Let G be a finite Abelian group. Given an integer a relatively prime to the order of G, let σ_a be the permutation on G defined by σ_a (x) = x^a. The problem studied in the paper is to determine the parity of σ_a. Generalizations of well-known theorems on the Legendre symbol are obtained in this context.     

A generalized notion of quotient for finite Abelian groups

A finite Abelian group G is partitioned into subsets which are translations of each other. A binary operation is defined on these sets in a way which generalizes the quotient group operation. Every finite Abelian group can be realized as such a generalized quotient with G cyclic.     

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

The Galois embedding problem is considered in the case of number fields and an elementary Abelian kernel. New cases are discovered, in which the consistency condition is sufficient for the existence of a solution of the embedding problem. In particular, it is true when the order of the kernel is the cube of a prime integer.     

Convolution functions on compact Abelian groups

ПустьG — бесконечная компактная абелева г руппа с группой характеров?, и дляr>0A _r (G) обозначает множест во всехf∈L ₁ (G), преобразование Фурь е которыхf принадлеж итl _r (?). Пусть, далее, дляr>0 иs>0A(r, s)(G) обозначает множество всех такихf∈L ₁ (G), чтоf принадлежит про странству ЛоренцаL(r,s)(?). Теорема 1. Пусть 1<р≦2, 1<q≦2и 1/r=1/р+ 1/q?1. ТогдаL _p (G)*L _q (G)?A _r ,(G), 1/r+1/r′=1, причем равенство имеет место в том и тол ько том случае, когда p=q=2. Теорема 2. Пусть p, q, r удовл етворяют условиям те оремы 1 и 1/s=1/p+1/q. Тогда

1. существуютf∈L _p (G) и h∈L_q(G) т акие, чтоf*h ? A(β,γ)(G) ни для какихβ0;
2. если 0<s ₀ , то существую тf ∈L _p (G) и h ∈L _q ,(G) такие, что f*h∈A(r′, s ₀)(G).

Из теоремы 2 следует, чт о неравенство Юнга в определенном смысле неулучшаемо. Непосредственными с ледствиями теоремы 2 я вляются также один результат Р. Л. Липсмана и теорема У. Б. Тевари—А. К. Гупта.