Maximal resolvability of totally bounded and ℵ0-bounded groups

The maximal resolvability of totally bounded groups (and, under the assumption that the generalized continuum hypothesis holds, of ₀-bounded groups) is proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 593–598, April, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-01374.     

Fredman’s reciprocity,invariants of abelian groups,and the permanent of the Cayley table

Let C_n{\mathcal{C}}_{n} denote the cyclic group of order n. For G=C_nG={\mathcal{C}}_{n}, we compute the Poincaré series of all C_n{\mathcal{C}}_{n}-isotypic components in (the symmetric tensor exterior algebra of ). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili–Jibladze. Then we consider the Cayley table, , of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of equals , where n is the order of G.     

Free groups in the smallest ideal of β G

Let G be an infinite group with cardinality κ embeddable into a direct sum of countable groups and let β G be the Stone-Čech compactification of G as a discrete semigroup. We show that the structural group of the smallest ideal of β G contains copies of the free group on generators. Supported by NRF grants FA2007041200005 and IFR2008041600015, respectively, and The John Knopfmacher Centre for Applicable Analysis and Number Theory.     

A theorem of Piccard’s type in abelian Polish groups

In the paper we prove a theorem of Piccard’s type which generalizes [9, Theorem 2]. More precisely, we show that in an abelian Polish group X the set \(\left\{ {\left( {{x_{1, \ldots ,\;}}{x_N}} \right) \in \;{X^N}\;:\;A\; \cap \;\bigcap\limits_{i = 1}^N {\left( {A + {x_i}} \right)} \;is\;not\;Haar\;meager\;in\;X} \right\}\) is a neighbourhood of 0 for every N ∈ N and every Borel non-Haar meager set A ? X. The paper refers to the paper [3].     

Gamma—rings and multiplications on abelian groups

If M and Γ are abelian groups, then M will be a Γ-ring iff there exists a group homomorphism f from Γ into the group of all multiplications of M, Mult(M), such that f(Γ) satisfies the Generalized Associativity Property on M. In this note we examine the following special cases of this result: (i) M is a Γ-ring satisfying the Nobusawa Condition, (ii) M is a cyclic group, (iii) M is a direct sum of cyclic groups and (iv) M is a Γ-ring that has unity elements.     

Virtually abelian Kähler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations.     

On the Erd?s-Ginzburg-Ziv constant of finite abelian groups of high rank

Let G be a finite abelian group. The Erd?s-Ginzburg-Ziv constant s(G) of G is defined as the smallest integer l∈N such that every sequence S over G of length |S|?l has a zero-sum subsequence T of length |T|=exp(G). If G has rank at most two, then the precise value of s(G) is known (for cyclic groups this is the theorem of Erd?s-Ginzburg-Ziv). Only very little is known for groups of higher rank. In the present paper, we focus on groups of the form , with n,r∈N and n?2, and we tackle the study of s(G) with a new approach, combining the direct problem with the associated inverse problem.     

Structure of augmentation quotients of finite homocyclic abelian groups

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.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(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.     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.     

On Σ-subsets of naturals over abelian groups

We obtain conditions for the Σ-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each e-ideal I there exists a torsion-free abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I; there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal Σ-function; for each principal e-ideal I there exists a periodic abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I.     

Torsion-free abelian α-irreducible groups of finite rank

If F is a free abelian group of finite rank and α is an endomorphism or an automorphism of its divisible hull, then the α‐ hull is determined, i.e. the minimal torsion-free abelian group with this endomorphism a. Torsion-free abelian groups of finite rank are called α-irreducible if their divisible hull is α-irreducible for an automorphism a. A complete classification is given for α-irreducible groups and this result is applied to groups of rank 2.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

On the automorphism groups of β-derived planes

This paper contains some general results about the automorphism group H of PG(3,q) leaving a -derived spread F invariant. The main result asserts that H contains a cyclic subgroups M^*, whose order divides (q+1)/2. In particular, H is not trivial. In terms of translation planes this means that the translation complement of any -derived plane always admits some collineations which are not dilatations.Research partially supported by M.P.I. (Research project Strutture Geometriche Combinatorie e loro Applicazioni).     

The signatures of the congruence subgroups G 0(τ) of the hecke groups G 4 and G 6

We determine the signatures of the congruence subgroups of the Hecke groups G ₄ and G ₆.     

Almost block independence and bernoullicity of ℤ d -actions by automorphisms of compact abelian groups

Summary We prove that a ^d -action by automorphisms of a compact, abelian group is Bernoulli if and only if it has completely positive entropy. The key ingredients of the proof are the extension of certain notions of asymptotic block independence from -actions to ^d -action and their equivalence with Bernoullicity, and a surprisingly close link between one of these asymptotic block independence properties for ^d -actions by automorphisms of compact, abelian groups and the product formula for valuations on global fields.Oblatum 20-X-1994     

On the critical pair theory in abelian groups: Beyond Chowla’s Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S + T| ≤ |S| + |T|. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rødseth and one of the authors.     

Fixed Points and Quartic Functional Equations in β-Banach Modules

Let M?=?{ 1, 2, . . . ,?n?} and let ${\mathcal {V}=\{\,I \subseteq M: 1 \in I\,\}}$ , where n is an integer greater than 1. Denote ${M{\setminus}{I}}$ by I ^c for ${I \in \mathcal {V}.}$ We investigate the solution of the following generalized quartic functional equation $$\begin{array}{ll} \sum\limits_{I \in\mathcal {V}}f\, \left({\sum\limits_{i \in I}}a_ix_i-\sum\limits_{i \in I^c}a_ix_i\right) \, = \,2^{n-2} \sum\limits_{1\leq i < j \leq n}a^2_{i}a^2_{j} \left[f(x_{i}+x_{j})+f(x_{i}-x_{j})\right] \\ \qquad \qquad \qquad \quad\quad\quad \quad\quad\quad\quad +\,2^{n-1} \sum\limits^{n}_{i=1}a^2_{i} \left(a^2_{i}-\sum\limits^{n}_{\substack{{j=1}\\{j\neq i}}}a^2_{j}\right)f(x_{i}) \end{array}$$ in β-Banach modules on a Banach algebra, where ${a_{1},\ldots, a_{n}\in \mathbb{Z}{\setminus}\{0\}}$ with a _? ?≠ ±1 for all ${\ell \in \{1 , 2, \ldots ,\,n-1\}}$ and a _n ?=?1. Moreover, using the fixed point method, we prove the generalized Hyers–Ulam stability of the above generalized quartic functional equation. Finally, we give an example that the generalized Hyers–Ulam stability does not work.