Periodic groups acting freely on Abelian groups

We describe {2, 3}-groups without elements of order 27, acting freely on an Abelian group. In particular, it is proved that such groups are locally finite.     

On groups acting freely on a tree

Suppose we are given a group G\mit\Gamma and a tree X on which G\mit\Gamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers a_d: = # {w ? vertX : w=gv, g ? G , d(v₀,w)=d }a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \} if d? ￥d\rightarrow \infty , where v, v_o are some fixed vertices in X.¶ In this paper we consider the case where G\mit\Gamma is a finitely generated group acting freely on a tree X. The growth function ?a_d x^d\textstyle\sum\limits a_d x^d is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where G\mit\Gamma is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.     

Characterizations of the Shunkov groups

We study the structure of a family of finite groups of the form L _g = 〈a, a ^g 〉 in a periodic Shunkov group. As a consequence of the obtained result, we get two characterizations of periodic Shunkov groups. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1110–1118, August, 2008.     

On Sylow subgroups of Shunkov periodic groups

We study the structure of Sylow 2-subgroups in Shunkov periodic groups with almost layer-finite normalizers of finite nontrivial subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1548–1556, November, 2005.     

On Shunkov groups with a strongly embedded subgroup

The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature.     

Shunkov groups saturated with general linear groups

We prove that if a periodic Shunkov group is saturated with degree 2 general linear groups over finite fields then it is isomorphic to the degree 2 general linear group over a suitable locally finite field.     

On the factorization of Abelian groups

In this note we consider a conjecture made by Professor de Bruijn and subsequent generalizations of this conjecture. We show by using constructions of his that in a certain sense these generalizations are the best possible. A survey of these known results is presented. Finally some new results of this kind are given.     

Periodic part in some Shunkov groups

A group G is saturated with groups in a set X if every finite subgroup of G is embeddable in G into a subgroup L isomorphic to some group in X. We show that a Shunkov group has a periodic part if the saturating set for it coincides with one of the following: {L₂(q)}, {Sz(q)}, {Re(q)}, or {U₃(2ⁿ)}. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 96–125, January–February, 1999.     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 121–126.Original Russian Text Copyright © 2005 by E. G. Sklyarenko.This revised version was published online in April 2005 with a corrected issue number.     

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module _E (G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if _R M is an arbitrary unitary left module and M ⁺ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M ⁺) under the canonical ring homomorphism R → E(M ⁺). Then it holds that if _E(M+) N is a pure submodule in _E(M+) M ⁺, then _R N is a pure submodule in _R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.     

On homogeneous stochastic processes on compact Abelian groups

This article discusses the extension to general compact Abelian groups of some results previously established by R. Roy for the case of the circle and the sphere. Estimators of the covariance function and spectral parameters for a homogeneous stochastic process defined on a compact Abelian group are considered and their properties are derived.     

On the product of finitely generated Abelian groups

It is shown that if a group G is a product of Abelian subgroups A and B one of which is finitely generated, then the group G will have a nontrivial normal subgroup that is contained either in A, or in B.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 443–446, March, 1973.     

On mixed Abelian groups with periodic groups of automorphisms

In the paper, sufficient conditions for the splittability of mixed Abelian groups with periodic automorphism groups are established. Classes of mixed splittable Abelian groups with perfect holomorphs are distinguished. Translated fromMaternaticheskie Zametki, Vol. 61, No. 4, pp. 483–493, April, 1997. Translated by A. I. Shtern