Abelian torsion groups with a countably compact group topology

Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants.We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than .We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κ^ω, for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ>κ.     

Abelian groups

The present, fourth survey of review articles on Abelian groups includes works reviewed in the years 1979–1984. Also, as in the preceding surveys, no attention has been given to questions involving finite Abelian groups, topological groups, ordered groups, group algebras, modules, or the structure of subgroups, or to questions connected with logic. The word group throughout is understood to mean Abelian group (except in the case of the group of automorphisms of an Abelian group). Concepts and notation not defined in this survey can be found in the books [117, 118]. The letterZ throughout denotes the group (or ring) of integers,Q the group (or field) of rational numbers,Q _p the group (ring) of all rational numbers whose denominator is not divisible by the primep, I_p the group of allp-adic integers. The torsion part of an Abelian groupG is throughout denoted bytG. Translated from Itogi Nauki i Tekhniki. Seriya Algebra, Topologiya, Geometriya, Vol. 23, pp. 51–118.     

Multidimensional Cooley–Tukey Algorithms Revisited

The representation theory of Abelian groups is used to obtain an algebraic divide-and-conquer algorithm for computing the finite Fourier transform. The algorithm computes the Fourier transform of a finite Abelian group in terms of the Fourier transforms of an arbitrary subgroup and its quotient. From this algebraic algorithm a procedure is derived for obtaining concrete factorizations of the Fourier transform matrix in terms of smaller Fourier transform matrices, diagonal multiplications, and permutations. For cyclic groups this gives as special cases the Cooley–Tukey and Good–Thomas algorithms. For groups with several generators, the procedure gives a variety of multidimensional Cooley–Tukey type algorithms. This method of designing multidimensional fast Fourier transform algorithms gives different data flow patterns from the standard “row–column” approaches. We present some experimental evidence that suggests that in hierarchical memory environments these data flows are more efficient.     

Isomorphism of a Ring to the Endomorphism Ring of an Abelian Group

This paper presents necessary and sufficient conditions under which isomorphism of endomorphism rings of additive groups of arbitrary associative rings with 1 implies isomorphism of these rings. For a certain class of Abelian groups, we present a criterion which shows when isomorphism of their endomorphism rings implies isomorphism of these groups. We demonstrate necessary and sufficient conditions under which an arbitrary ring is the endomorphism ring of an Abelian group. This solves Problem 84 in L. Fuchs’ “Infinite Abelian Groups.”__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 231–234, 2003.     

On Groups with Minimality Condition for Noninvariant Abelian pd-Subgroups

We study properties and the structure of non-Abelian groups with minimality condition for non-invariant Abelian pd-subgroups in the case where they do not satisfy the minimality condition for Abelian pd-subgroups. We prove the solvability of these groups and establish relations with non-Abelian groups in which all infinite Abelian pd-subgroups are invariant.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 265–270, February, 2005.     

Topological groups with the coalgebraic lattice of closed subgroups

Locally compact groups with a coalgebraic lattice of closed subgroups are investigated and Abelian and zero-dimensional nilpotent groups with a coalgebraic lattice of closed subgroups are described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 7 and 8, pp. 1102–1107, July–August, 1991.     

Torsion-free Abelian groups with cyclic p-basic subgroups

We investigate the structure of indecomposable torsion-free Abelian groups all of whose p-basic subgroups are cyclic, and also the structure of the groups and rings of endomorphisms of such groups. We prove the existence of a torsion-free Abelian group of countable rank with cyclic p-basic subgroups which has no indecomposable nonzero direct summands.Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 805–813, December, 1976.     

On noncommutative serving extensions of groups

The paper studies serving extensions of noncommutative groups in the sense of the definition transferred naturally from the theory of Abelian groups. The basic results are those continuing the Los Theorem on the de composability of Abelian serving extensions of algebraic compact groups in the category of all groups.Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 283–291, March, 1972.The author wishes to take this opportunity to thank her scientific adviser Prof. L. Ya. Kulikov for his attention to this work.     

Complete direct sums of Abelian torsion-free groups of rank 1 with isomorphic groups or rings of endomorphisms. I

Isomorphism conditions are established for torsion-free Abelian groups with isomorphic groups or rings of endomorphisms decomposable into complete direct sums of groups of rank 1.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 867–878, December, 1973.     

Group Extensions and Hall Polynomials

In this paper, we give two proofs of a formula containing the numbers of automorphisms of an Abelian group, of its subgroups, and of its quotient groups. The first proof is based on the use of the theory of Hall polynomials, while the second one uses extension theory for Abelian groups.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 180–185.Original Russian Text Copyright © 2005 by G. V. Voskresenskaya.     

On varieties of n-solvable groups

A new and shorter proof is given for the fundamental theorem of Kaluzhnin's recent note [1]: A variety of n-Abelian groups is the sum of a variety of Abelian groups and the respective Burnside varieties of groups of exponent n and 1–n. This theorem is extended to varieties of n-solvable groups.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 629–634, December, 1968.     

Definability of vector groups by endomorphism semigroups

Some conditions for torsion-free Abelian groups to be isomorphic are found; the groups in question are decomposable into a direct product of rank 1 groups with isomorphic endomorphism semigroups.Translated fromAlgebra i Logika, Vol. 33, No. 4, pp. 422–428, July-August, 1994.     

Topological groups with the generalized minimality condition for subgroups

Locally compact, locally solvable groups with the generalized minimality condition for closed Abelian subgroups are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1394–1402, October, 1990.     

Groups all of whose infinite abelian pd-subgroups are normal

The author studies groups in which any infinite Abelian pd-subgroup (p is a prime) is normal, on the assumption that the group indeed contains such subgroups (IH_p-groups). Necessary and sufficient conditions are established for a group to be an IH_p-group. Relationships are established between the class of IH_p-groups and the class of groups in which all infinite Abelian subgroups are normal, and the class of groups in which all pd-subgroups are normal.Translated from Ukrainskii Matematicheskii Zhurnal, Vo. 44, No. 6, pp. 796–800, June, 1992.     

Elementary theories of complete torsion-free Abelian groups with p-ary topology

The complete torsion-free Abelian groups are investigated as dibasic RPC models whose second basic set comprises subgroups forming a chain with factors of order p. Two theorems about the elementary theories of classes of such models are proved.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 201–208, August, 1973.     

Definability of separable torsion-free Abelian groups by endomorphism semigroups

In the class of separable torsion-free Abelian groups, we describe those which are defined by their endomorphism semigroups.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 523–530, September-October, 1995.     

Abelian p-extensions of an absolutely unramified field

In this paper the characteristic of the Abelian p-extensions of a field is given using valuations from its cyclotomic extensions. On the basis of this the character group of the Galois groups of the maximal Abelian p-extension of an absolutely unramified complete field is calculated. The results are applied to the case of a two-dimensional local field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 80–90, 1991.     

Torsion of Abelian varieties of finite characteristic

The finiteness of the torsion of Abelian varieties with a complete real field of endomorphisms in the maximal Abelian extension of the field of definition is proven. This assertion is formally deduced from the finiteness hypothesis for isogenic Abelian varieties, proven for characteristic p > 2. The structure is studied of the Lie algebra of Galois groups acting in a Tate module; in particular, for fields of characteristic greater than two there is proven one-dimensionality of the center of the Lie algebra.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 3–11, July, 1977.     

Abelian varieties in characteristic p

In this paper Tate's finiteness conjecture for isogenies of polarized Abelian varieties in characteristic p>2 is proved. From this conjecture it is deduced that Tate modules are semisimple and that Tate's conjecture on the homomorphisms of Abelian varieties is valid.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 393–400, March, 1976.     

Periodic T(Ā)-groups with Abelian locally nilpotent coradical

The article describes periodic groups in which non-Abelian normal divisors satisfy the transitivity condition and which have an Abelian locally nilpotent coradical."Gorsistemotekhnika" Scientific-Industrial Association. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 81–85, 1992;