On purifiable torsion-free rank-one subgroups

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

Two operators or two subgroups of a group will be called I-conjugate whenever they are conjugate under the group of isomorphisms of this group. It is well known that all the cyclic subgroups of highest order contained in any abelian group are I-conjugate. A necessary and sufficient condition that all the cyclic subgroups of any other order contained in an abelian group G of order pmm, p being a prime number, are I-conjugate is that all the invariants of G are equal to each other. When this condition is satisfied we shall prove, in particular, that the number of the subgroups of order pa, a c m , contained in G is equal to the number of its subgroups of index pa. It is known that the ?-subgroup of G is composed of the pth power of every operator of G. The concept of +-subgroup in connection with abelian groups can readily be extended by considering the characteristic subgroup of G composed of the path power of every operator of G. This subgroup may be called the Oa-subgroup of G. It is composed of all the operators of G which are found in each one of its subgroups of index pa, and all the operators of highest order in a 4.-subgroup of G are I-conjugate. In particular, the 0&-subgroup generated by operators of order p is the fundamental characteristic subgroup of G. t In fact, when p > 2 the 0a-subgroup which is generated by operators of order pr is contained in every characteristic subgroup of G which involves operators of order pr. The quotient group of G corresponding to a 4a-subgroup may be called a 0,fi-quotient group. This quotient group is simply isomorphic with the subgroup of G generated bv all its operators whose orders divide p> Hence the following: THEOREM. The number of the subgroups of index pa contained in any abelian

Algebraic structure of codes over Fq, closed under arbitrary abelian group G of permutations with exponent relatively prime to q, called G-invariant codes, is investigated using a transform domain approach. In particular, this general approach unveils algebraic structure of quasi-cyclic codes, abelian codes, cyclic codes, and quasi-abelian codes with restriction on G to appropriate special cases. Dual codes of G-invariant codes and self-dual G-invariant codes are characterized. The number of G-invariant self-dual codes for any abelian group G is found. In particular, this gives the number of self-dual l-quasi-cyclic codes of length ml over Fq when (m,q)=1. We extend Tanner's approach for getting a bound on the minimum distance from a set of parity check equations over an extension field and outline how it can be used to get a minimum distance bound for a G-invariant code. Karlin's decoding algorithm for a systematic quasi-cyclic code with a single row of circulants in the generator matrix is extended to the case of systematic quasi-abelian codes. In particular, this can be used to decode systematic quasi-cyclic codes with columns of parity circulants in the generator matrix.

We prove that any finitely generated torsion free solvable subgroup of the group ${\rm IET}$ of all Interval Exchange Transformations is virtually abelian. In contrast, the lamplighter groups $A\wr \mathbb{Z}^k$ embed in ${\rm IET}$ for every finite abelian group $A$, and we construct uncountably many non pairwise isomorphic 3-step solvable subgroups of ${\rm IET}$ as semi-direct products of a lamplighter group with an abelian group. We also prove that for every non-abelian finite group $F$, the group $F\wr \mathbb{Z}^k$ does not embed in ${\rm IET}$.

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G byHi/Hi−1 is the subgroup of all FC elements in G/Hi−1.

In the paper, necessary and sufficient conditions for an Abelian group A to be isomorphic to the endomorphism group End(A) are obtained. The classes of periodic Abelian groups, divisible Abelian groups, nonreduced Abelian groups, and reduced algebraically compact Abelian groups are considered. For certain classes of Abelian groups, the isomorphism problem for a group and its endomorphism group is solved under the assumption that the endomorphism group itself has the corresponding property.

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability of the quadratic equation is also established on the noncommutative group $T(2, K)$, where $K$ is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

The general solution of the multiplicative symmetric functional equation $$f(f(x)y)=f(xf(y))$$, in abelian groups, was given by Dhombres. But it is still an open problem in non-abelian groups and abelian quasi-groups. In this paper, by using a new approach introduced by the author, we give a vast class of its solutions in arbitrary semigroups. It also works for non-abelian groups, and gives its general solution in abelian groups with a different representation from Dhombres’s solution. Also, we introduce some related equations and closed relationships between decomposer type and multiplicative symmetric equations. It is worth noting that the new approach and methods can be used for studying the equation on magmas (groupoids) as the most general algebraic structures for the topic. At last we pose a conjecture that guesses the general solution in arbitrary groups (and possibly semigroups).

Some results from [2], [5], [6] are generalized for finite automata over arbitrary groups. The accepting power is smaller when abelian groups are considered, in comparison with the non-abelian groups. We prove that this is due to the commutativity. Each language accepted by a finite automaton over an abelian group is actually a unordered vector language. Finally, deterministic finite automata over groups are investigated.

Let C be an Abelian group. An Abelian group A from a class X of Abelian groups is said to be CH-definable in X if, for any group B ∈ X, the isomorphism Hom(C,A) ≅ Hom(C,B) implies that A ≅ B. If every group from X is CH-definable in X, then X is called an CH-class. In this paper, we study conditions under which a class of completely decomposable torsion-free Abelian groups is an CH-class, where C is a vector group.

The failure of spectral synthesis on some types of discrete Abelian groups

We consider the Kac–Bernstein functional equation $$\begin{aligned} f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \quad x, y\in X, \end{aligned}$$ on an arbitrary Abelian group X. We give a complete description of the solutions of this equation in the class of positive functions. This result is a generalization of a theorem by Pl. Kannappan. We also study the solutions of this equation in the class of complex-valued Hermitian functions.

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.

Cohomological dimension and acyclic resolutions

Let G be an Abelian group of order n, let k 2 be an integer, and A1, . . ., A k be non-empty subsets of G.and maximal by inclusion if for any i {1, . . ., k} and x G \ Ai, the collection (A1, . . ., Ai-1, Ai {x}, Ai+1, . .In this work, we study the problem of the maximal value of k (G).In particular, the maximal value of k (Z d ) for the cyclic group Z d is determined.Upper and lower bounds for k (G) are obtained for the Abelian group G.The structure of maximal k-sum-free set by capacity (by inclusion) is described for an arbitrary cyclic group.