On characterized subgroups of compact abelian groups

Topology and its Applications

et al. 2017

MSC: Keywords:Characterized subgroup a-sequence of integers Circle group Dirichlet set Arbault set Factorizable subgroup A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u = (u n ) n∈N0 in N such that u n x → 0 uniformly on A. In particular, A is contained in the subgroup t u (T) := {x ∈ T : u n x → 0}, which is the subgroup of T characterized by u. Using strictly increasing sequences u in N such that u n divides u n+1 for every n ∈ N, we find in T a family of closed perfect D-sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D-sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Finally, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.

Section: A-set)mentioning

confidence: 99%

Dirichlet sets vs characterized subgroups

Barbieri

Topology and its Applications

et al. 2017

“…On the other hand, if an abelian group G has no divisible quotients, then it has no divisible subgroups. To complete this comment on Proposition 2.7, we add that abelian groups G with no divisible quotients were characterized in [6] for example as those abelian groups admitting no surjective homomorphism on Z(p ∞ ) for any prime p, where Z(p ∞ ) denotes the Prüfer group.…”

Section: W-divisible Groupsmentioning

confidence: 99%

A Factorization Theorem for Topological Abelian Groups

Communications in Algebra

2014

“…To completely determine all almost orthogonal pairs, it is relevant to know when a given group G admits divisible quotients and which divisible groups can be obtained as quotients of G. The groups that do not admit a divisible quotient were described in [8]. For the sake of completeness, and due to the fact that [8] is not easily accessible, we provide a proof of this theorem, formulated in a counterpositive form that is more appropriate for our purposes. (b) Some non-trivial quotient of G is divisible.…”

Section: Fully Inert Subgroups Of Divisible Groupsmentioning

confidence: 99%

“…Using this fact, the following further condition, equivalent to those of Theorem 3.5, was given in [8]: there exists a subgroup N of G such that G has no maximal proper subgroups containing N .…”

Section: Fully Inert Subgroups Of Divisible Groupsmentioning

confidence: 99%

Fully inert subgroups of divisible Abelian groups

Journal of Group Theory

Salce

et al. 2013

Abstract.A subgroup H of an Abelian group G is said to be fully inert if the quotient .H C .H //=H is finite for every endomorphism of G. Clearly, this is a common generalization of the notions of fully invariant, finite and finite-index subgroups. We investigate the fully inert subgroups of divisible Abelian groups, and in particular, those Abelian groups that are fully inert in their divisible hull, called inert groups. We prove that the inert torsion-free groups coincide with the completely decomposable homogeneous groups of finite rank and we give a complete description of the inert groups in the general case. This yields a characterization of the fully inert subgroups of divisible Abelian groups.