Enumeration theorems in infinite Abelian groups

Bounded subgroups as a von Neumann radical of an Abelian group

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

Minimal sumsets in finite solvable groups

Let A be a weakly sequentially complete Banach algebra containing a bounded approximate identity that is an ideal in its second dual A∗∗. We refer to such an algebra as a WeSeBai algebra. In the present paper, we examine the Arens regularity properties of closed ideals of algebras in this class and observe that, although WeSeBai algebras themselves are always strongly Arens irregular, a variety of Arens regularity properties can be found among their closed ideals. After characterizing Arens regular ideals and strongly Arens irregular ideals, we focus on the main examples of WeSeBai algebras: the convolution group algebras L1(G), G compact, and the Fourier algebras A(Γ), Γ discrete and amenable. We find examples of Arens regular ideals in L1(G) and A(Γ), reflexive and nonreflexive, and examples of strongly Arens irregular ideals that are not in the WeSeBai class. For this, we introduce a new class of Riesz sets that are not Λ(p), for any p>1, and show how to obtain them in a broad family of noncommutative groups. As a consequence of our approach, we prove that every infinite Abelian group contains a Rosenthal set that is not Λ(p), for any p>0.

The P-DNP Problem for Infinite Abelian Groups

We formulate U(1)bda Chern-Simons theory, which is also called BF theory, on a lattice, adapting a method proposed by Kantor and Susskind [1] for the groups R and ZN. Our method applies to any finite or infinite abelian group. We study the discrete symmetries and use the model to provide a rigorous treatment of the composite fermion theory of the fractional quantum Hall effect (FQHE) [2][3][4], with no ambiguities relating to intersecting Wilson/'t Hooft lines. We derive Jain's fractions, and one can also calculate corrections to the mean field solution within this framework. We also generalize the formalism to higher form gauge models in arbitrary dimension, and suggest a possible non-Abelian extension.

We present a general algebraic basis for arbitrary systems of units such as those used in physical sciences, engineering, and economics. Physical quantities are represented as q-numbers: an ordered pair u = {u,labelu}, that is, u ∈ q = X × W B . The algebraic structure of the infinite sets of labels that represent the “units” has been established: such sets W B are infinite Abelian multiplicative groups with a finite basis. W B is solvable as it admits a tower of Abelian subgroups. Extensions to include the possibility of rational powers of labels have been included, as well as the addition of named labels. Named labels are an essential feature of all practical systems of units. Furthermore, q is an Abelian multiplicative group, and it is not a ring. q admits decomposition into one-dimensional normed vector spaces over the field X among members with equivalent labels. These properties lead naturally to the concept of well-posed relations, and to Buckingham’s theorem of dimensional analysis. Finally, a connection is made with a Group Ring structure and an interpretation in terms of the observable properties of physico-chemical systems is given.

Combinatorial Conditions Forcing Commutativity of an Infinite Group

В наших предыдущих работах мы продемонстрировали, что теория конечных подмножеств различных ассоциативных алгебр позволяет интерпретировать элементарную арифметику, в частности, она неразрешима. Например, это было показано для любых бесконечных абелевых групп. Возникает естественный вопрос: можно ли обобщить этот результат на более широкий класс алгебр, скажем, все коммутативные моноиды. В некоторых случаях нами ответ тоже получен ранее: для коммутативных моноидов с сокращением, имеющим элемент бесконечного порядка или произвольных абелевых групп. Сейчас же мы продемонстрируем, что не для всяких коммутативных моноидов это верно. Более того, мы дадим описание конструкции, которая позволяет строить такого рода системы из разного рода исходных алгебр. Вместе с тем, будут указаны и некоторые границы её применимости. In our previous works, we have proved for various associative algebras that the finite subsets theory allows to interpret elementary arithmetic, in particular, such theory is undecidable. For example, this is proved for all infinite Abelian groups. A natural question arises: can we generalize this result to a wider class of algebras, for example, all commutative monoids. In some cases, we also have proved analogous result: for commutative cancellative monoids with an element of infinite order, or arbitrary Abelian groups. In this paper we prove that this is not true for arbitrary commutative monoids. Moreover, we propose a method that allows to construct such algebras by various original algebras. Also, we have found a limitation of this method.

We show that if a countable group G G is the free product of infinite abelian groups, then for every free, probability-measure-preserving (p.m.p.) action of G G , its orbit equivalence class is weakly dense in the space of p.m.p. actions of G G . This extends Lewis Bowen’s result for free groups.

We describe the characteristic subgroups of a transitive abelian 2-group, thus completing Kaplansky's study in section 18 of his classic monograph Infinite abelian groups. This appears to be a new result even for finite abelian 2-groups.

We give a model-theoretic proof of the fact that for all infinite Abelian groups P ≠ NP in the sense of binary nondeterminism. This result has been announced 1994 by Christine Gaßner.

A set of elements in which a rule of combination o is so defined as to satisfy the following three postulates shall be called an Abelian group with respect to 0: 1) a o b-b o a, whenever a, b and b o a belong to the set. 2) (aob)oc= ao(boc), whenever a, b, c, aob, boc and ao(boc) belong to the set. 3) For every two elements a and b (a = b or a $ b) there is an element x in the set such that a o x = b. If we wish to distinguish between finite and infinite groups we may add a fourth postulate, either a) The set contains n elements; or b) The set is infinite.

On supersimple groups

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group.