可除阿贝尔p-群的自同构群

主要探讨了秩大于或者等于p-1的可除阿贝尔p-群的p-自同构群,并且得到这些p-自同构如何作用在该可除阿贝尔p-群上.这些结论有助于进一步理解Cernikov p-群的结构.     

Group algebras of generalized primary groups

A number of the known results concerning group algebras of primary groups carry over to group algebras of generalized primary groups. In particular, we show that the group algebra LG of a generalized primary (relative to the prime p) group G over the ring L, in which the element p is not invertible, determines, to within an isomorphism, a basis subgroup of the generalized primary group G. In addition, we indicate two classes of composite abelian groups which are determined, to within an isomorphism, by their group algebras over the ring L.     

Nilpotency of the multiplicative group of a group ring

It is proven that if K is a commutative ring of characteristic p^m while group G contains p-elements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G is a finite p-group. Those group algebras KG are described for which the nilpotency classes of groups G and UKG coincide.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 191–200, February, 1972.In conclusion, the author wishes to express her gratitude to A. A. Bovdi for his scientific direction.     

The ring of integers of an Abelian extension of an algebraic number field as a Galois module

The ringO of integers of a finite Abelian extension K of an algebraic number field k is studied as a module over the group ring =[G], where is the ring of integers of k and G is the Galois group of K/k. It is proved that the ring is a decomposable -module if and only if there exists in K/k an intermediate extension K/F. FK, whose degree divides the different.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta. im. V. A. Steklova AN SSSR, Vol. 71, pp. 80–84, 1977.     

Compactness conditions for groups of automorphisms of topological groups

It is proved that if G is a compact, totally disconnected Abelian group and Aut G is its group of topological automorphisms (with the natural topology), then the following conditions are equivalent: (a) Aut G is compact; (b) Aut G is locally compact; (c) Aut G has small invariant neighborhoods of the identity; (d) Aut G is an -group; (e) the factor group of Aut G by its center is compact; (f) the closure of the commutator subgroup of Aut G is compact; (g) , where F_p is a finite p-group, Z_p is the additive group of p-adic integers, and n_p < .Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 735–743, May, 1976.In conclusion, the author thanks V. P. Platonov for his constant attention to this paper.     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.     

On a conjecture of Herzer concerning linear partitions

We prove: Let G be a finite group with a non-trivial partition (G) and a set (G) of subgroups such that any two components of (G) are contained in at least one W with W (G) and W G; then G is a p-group or a Frobenius group.From this we obtain: A finite group with a linear partition can only be an abelian p-group or a Frobenius group.Dedicated to Prof. H. Lenz on the occasion of his 70^th birthday     

p-Groups of automorphisms of abelianp-groups

We consider the action of a p-group G on an Abelian p-group A, with the latter treated as a faithful right ℤG-module. Our aim is to establish a connection between exponents of the kernels under the induced action of G on elementary p-groups A/pA and Ω₁(A) = {x ε A|px=0}; the kernels are denoted by C_G(A/pA) and C_G(Ω₁(A)), respectively. It is proved that if the exponent of one of the kernels C_G(A/pA) or C_G(Ω₁(A)) is finite then the other also has a finite exponent bounded in terms of the first; moreover, these kernels are nilpotent. In one case we impose the additional restriction . And the wreath product of a quasicyclic group and an arbitrary p-group G shows that this condition cannot be dropped. The results obtained are used to confirm, for one particular case, the conjecture on the boundedness of a derived length of a finite group with an automorphism of order 2 all of whose fixed points are central. (The solubility of such groups, and also the reduction to the case of 2-groups, were established in [1].) Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 359–371, May–June, 2000.     

Circle and Line Bundles Over Generalized Weyl Algebras

Strongly \(\mathbb {Z}\)-graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras \({\mathcal B}(p;q,0)\) (over a ring of polynomials in one variable) are constructed. The Chern-Connes pairing between the cyclic cohomology of \({\mathcal B}(p;q,0)\) and the isomorphism classes of sections of associated line bundles over \({\mathcal B}(p;q,0)\) is computed, thus demonstrating that these bundles, which are labelled by integers, are non-trivial and mutually non-isomorphic. The constructed strongly \(\mathbb {Z}\)-graded algebras are shown to have Hochschild cohomology reminiscent of that of Calabi-Yau algebras. The paper is supplemented by an observation that a grading by an Abelian group in the middle of a short exact sequence is strong if and only if the induced gradings by the outer groups in the sequence are strong.     

The enumeration of subgroups in finite p-groups

We generalize the familiar principle of enumeration due to Hall and establish a new principle for the enumeration of subgroups of any p-group G of order p^m, based on the following grouptheoretic relation found by the author: $$\sum\nolimits_{\lambda = 0}^m {\left( { - 1} \right)^\lambda p^{\left( {\begin{array}{*{20}c} \lambda \\ 2 \\ \end{array} } \right)} \mathcal{E}_\lambda \left( G \right)} = 0,$$ where ?_λ (G) is the number of elementary Abelian subgroups of order p^λ in G.     

On groups with isomorphic holomorphs

Groups with isomorphic holomorphs are said to be holomorphically isomorphic. The following problems are treated.

1. What Abelian groups have the property that their holomorphic isomorphism implies the isomorphism of the groups themselves?
2. Does holomorphic isomorphism of two groups, one of which is Abelian, imply the commutativity of the other group?

Classes of groups are selected for which the answers to these questions are positive.     

The k-Orbit Reconstruction for Abelian and Hamiltonian Groups

Let G be a permutation group on a set . Then G acts in the natural way on the collection ^{k} of all k-element subsets. Orbits under this action are called k-orbits. A problem similar to the Edge-Reconstruction Conjecture in graph theory can be posed for k-orbits of a general group G. Here the k-orbit reconstruction problem is solved for transitive Abelian and Hamiltonian groups: all k-orbits of Abelian groups are reconstructible if k>3 and the same is true for Hamiltonian groups if k>4.     

The quasivariety generated by a torsion-free Abelian-by-finite group

Let L_q(qG) be the quasivariety lattice contained in a quasivariety generated by a group G. It is proved that if G is a finitely generated torsion-free group in (i.e., G is an extension of an Abelian group by a group of exponent 2ⁿ), which is a split extension of an Abelian group by a cyclic group, then the lattice L_q(qG) is a finite chain. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 407–427, July–August, 2007.     

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.     

Nonreduced Abelian groups with UA-rings of endomorphisms

A ring K is a unique addition ring (a UA-ring) if its multiplicative semigroup (K, · ) can be equipped with a unique binary operation + transforming this semigroup to a ring (K, ·, +). An Abelian group is called an End-UA-group if its endomorphism ring is a UA-ring. In the paper, we find End-UA-groups in the class of nonreduced Abelian groups.     

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.     

内交换p-群的中心扩张(I)

设N,H为群. H被N的中心扩张是关于群的短正合序列:满足N≤Z(G).本文完全分类了当|N|=p,H为内交换p-群时, H被N的中心扩张所得的群.       

Discretely normed Abelian groups

A discrete norm on an Abelian groupA is a non-negative function · A which satisfies the triangle inequality, is homogenous with respect to scaling ofA by and is bounded away from 0 onA/{0}.A countable Abelian group is discretely normed if and only if the group is free.