Displaying similar documents to “The dual group of a dense subgroup”

Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort, Lewis C. Robertson

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For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly...

Metrization criteria for compact groups in terms of their dense subgroups

Dikran Dikranjan, Dmitri Shakhmatov (2013)

Fundamenta Mathematicae

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According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G δ -dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined...

Cellularity and the index of narrowness in topological groups

Mihail G. Tkachenko (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study relations between the cellularity and index of narrowness in topological groups and their G δ -modifications. We show, in particular, that the inequalities in ( ( H ) τ ) ≤ 2 τ · in ( H ) and c ( ( H ) τ ) ≤ 2 2 τ · in ( H ) hold for every topological group H and every cardinal τ ≥ ω , where ( H ) τ denotes the underlying group H endowed with the G τ -modification of the original topology of H and in ( H ) is the index of narrowness of the group H . Also, we find some bounds for the complexity of continuous real-valued functions f on an arbitrary ω -narrow group...

A note on a class of factorized p -groups

Enrico Jabara (2005)

Czechoslovak Mathematical Journal

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In this note we study finite p -groups G = A B admitting a factorization by an Abelian subgroup A and a subgroup B . As a consequence of our results we prove that if B contains an Abelian subgroup of index p n - 1 then G has derived length at most 2 n .

Subgroups of ℝ -factorizable groups

Constancio Hernández, Mihail G. Tkachenko (1998)

Commentationes Mathematicae Universitatis Carolinae

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The properties of ℝ -factorizable groups and their subgroups are studied. We show that a locally compact group G is ℝ -factorizable if and only if G is σ -compact. It is proved that a subgroup H of an ℝ -factorizable group G is ℝ -factorizable if and only if H is z -embedded in G . Therefore, a subgroup of an ℝ -factorizable group need not be ℝ -factorizable, and we present a method for constructing non- ℝ -factorizable dense subgroups of a special class of ℝ -factorizable groups. Finally, we construct...

The Hughes subgroup

Robert Bryce (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let G be a group and p a prime. The subgroup generated by the elements of order different from p is called the Hughes subgroup for exponent p . Hughes [3] made the following conjecture: if H p ⁢ G is non-trivial, its index in G is at most p . There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case p = 3 and G arbitrary, and that of Hogan and Kappe [2] concerning the case when G is metabelian, and p arbitrary. A common...