Locally compact groups with complementary closed subgroups

Topological groups with complementary closed abelian subgroups and with complementary monothetic subgroups are considered.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 641–649, December, 1969.     

Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and a (not necessarily continuous) character of H. Then there exists a sequence of (continuous) characters of G such that limn→∞χn(α)=f(α) for all α∈H and does not converge whenever α∈G?H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.     

Limits in compact Abelian groups

For X a compact Abelian group and B an infinite subset of its dual , let CB be the set of all x∈X such that converges to 1. If F is a free filter on , let . The sets CB and DF are subgroups of X. CB always has Haar measure 0, while the measure of DF depends on F. We show that there is a filter F such that DF has measure 0 but is not contained in any CB. This generalizes previous results for the special case where X is the circle group.     

Tight subgroups in torsion-free Abelian groups

LetX be a torsion-free abelian group. We study the class of all completely decomposable subgroups ofX which are maximal with respect to inclusion. These groups are called tight subgroups ofX and we state sufficient conditions on a subgroup to be tight. In particular we consider tight subgroups of bounded completely decomposable groups. For those we show that every regulating subgroup is tight and we characterize the tight subgroups of finite index in almost completely decomposable groups. The second author was supported by a MINERVA fellowship.     

Abelian Carter subgroups in finite permutation groups

We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

On projective invariant subgroups of Abelian groups

The properties of projective invariant subgroups are studied. The structure of these subgroups in nonreduced groups is described. The conditions under which projective invariant subgroups are fully invariant are considered.     

Convolution functions on compact Abelian groups

ПустьG — бесконечная компактная абелева г руппа с группой характеров?, и дляr>0A _r (G) обозначает множест во всехf∈L ₁ (G), преобразование Фурь е которыхf принадлеж итl _r (?). Пусть, далее, дляr>0 иs>0A(r, s)(G) обозначает множество всех такихf∈L ₁ (G), чтоf принадлежит про странству ЛоренцаL(r,s)(?). Теорема 1. Пусть 1<р≦2, 1<q≦2и 1/r=1/р+ 1/q?1. ТогдаL _p (G)*L _q (G)?A _r ,(G), 1/r+1/r′=1, причем равенство имеет место в том и тол ько том случае, когда p=q=2. Теорема 2. Пусть p, q, r удовл етворяют условиям те оремы 1 и 1/s=1/p+1/q. Тогда

1. существуютf∈L _p (G) и h∈L_q(G) т акие, чтоf*h ? A(β,γ)(G) ни для какихβ0;
2. если 0<s ₀ , то существую тf ∈L _p (G) и h ∈L _q ,(G) такие, что f*h∈A(r′, s ₀)(G).

Из теоремы 2 следует, чт о неравенство Юнга в определенном смысле неулучшаемо. Непосредственными с ледствиями теоремы 2 я вляются также один результат Р. Л. Липсмана и теорема У. Б. Тевари—А. К. Гупта.     

Fully inert subgroups of free Abelian groups

A subgroup \(H\) of an Abelian group \(G\) is called fully inert if \((\phi H + H)/H\) is finite for every \(\phi \in \mathrm{End}(G)\) . Fully inert subgroups of free Abelian groups are characterized. It is proved that \(H\) is fully inert in the free group \(G\) if and only if it is commensurable with \(n G\) for some \(n \ge 0\) , that is, \((H + nG)/H\) and \((H + nG)/nG\) are both finite. From this fact we derive a more structural characterization of fully inert subgroups \(H\) of free groups \(G\) , in terms of the Ulm–Kaplansky invariants of \(G/H\) and the Hill–Megibben invariants of the exact sequence \(0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0\) .