The number of non-solutions of an equation in a group

Klyachko, Anton A.; Trofimov, A. V.

doi:10.1515/jgth.2005.8.6.747

Cited by 15 publications

(5 citation statements)

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“…(a) Ol ′ shanskij's example [28] of a countable non-topologizable group has discrete Z G . (b) Klyachko and Trofimov [23] constructed a finitely generated torsion-free group G such that Z G is discrete. (c) Trofimov [32] proved that every group H admits an embedding into a group G with discrete Z G .…”

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confidence: 99%

The Markov–Zariski topology of an abelian group

Dikranjan

Shakhmatov

2010

Journal of Algebra

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According to Markov (1946) [24], a subset of an abelian group G of the form {x ∈ G: nx = a}, for some integer n and some element a ∈ G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally bounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian T 1 topology Z G on G called the Zariski, or verbal, topology of G; see Bryant (1977) [3]. We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Fréchet-Urysohn.For a countable family F of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology T on G such that the T -closure of each member of F coincides with its Z G -closure. As an application, we provide a characterization of the subsets of G that are Tdense in some Hausdorff group topology T on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a longstanding problem of Markov (1946) [24]. We use P and N to denote the sets of all prime numbers and all natural numbers, respectively. In this paper, 0 ∈ N. As usual, Z denotes the group of integers, and Z(n) denotes the cyclic group of order n. We use c to denote the cardinality of the continuum. The symbol ω 1 denotes the first uncountable cardinal.

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confidence: 99%

The Markov–Zariski topology of an abelian group

Dikranjan

Shakhmatov

2010

Journal of Algebra

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“…(ii)→(i) follows from Lemma 5.6(b) and Theorem 2.7. According to [5] there exists a countable torsion-free non-topologizable group G. Hence for this group no cyclic subgroup C is Markov embedded into G by Lemma 5.6. The next proposition yields that none of them is a normal subgroup of G.…”

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confidence: 99%

Reflection principle characterizing groups in which unconditionally closed sets are algebraic

Dikranjan¹,

Shakhmatov²

2008

Journal of Group Theory

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We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63 years old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G.According to Markov [6], a subset S of a group G is called:(a) elementary algebraic if there exist an integer n > 0, a 1 , . . . , a n ∈ G and ε 1 , . . . , ε n ∈ {−1, 1} such that S = {x ∈ G : x ε 1 a 1 x ε 2 a 2 . . . a n−1 x εn = a n }, (b) algebraic if S is an intersection of finite unions of elementary algebraic subsets of G, (c) unconditionally closed if S is closed in every Hausdorff group topology of G.Since the family of all finite unions of elementary algebraic subsets of G is closed under finite unions and contains all finite sets, it is a base of closed sets of some T 1 topology Z G on G, called the Zariski topology of G. (This topology is also known under the name verbal topology, see [1].) The family of all unconditionally closed subsets of G coincides with the family of closed subsets of a T 1 topology M G on G, namely the infimum (taken in the lattice of all topologies on G) of all Hausdorff group topologies on G.

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“…In [20] (see also [18]), Markov proved the coincidence of unconditionally closed sets with algebraic sets for countable groups; a general (negative) answer was given only in 1979 by Hesse [12], who constructed an example of an uncountable non-topologizable group in which the complement to the identity element is not algebraic. The first countable non-topologizable group was given in 1980 by Ol'shanskii [21]; other examples were constructed in [17] and [16]. In contrast to these negative results, Zelenyuk [27] proved that each infinite group G admits a non-discrete regular topology with continuous shifts and continuous inversion.…”

Section: Problem 12 Find Necessary and Sufficient Conditions For Thmentioning

confidence: 99%

Topologizations of a set endowed with an action of a monoid

Банах

Протасов

Sipacheva

2014

Topology and its Applications

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Abstract. Given a set X and a family G of self-maps of X, we study the problem of the existence of a nondiscrete Hausdorff topology on X with respect to which all functions f ∈ G are continuous. A topology on X with this property is called a G-topology. The answer is given in terms of the Zariski G-topology ζ G on X, that is, the topology generated by the subbase consisting of the sets {x ∈ X : f (x) = g(x)} and {x ∈ X : f (x) = c}, where f, g ∈ G and c ∈ X. We prove that, for a countable monoid G ⊂ X X , X admits a non-discrete Hausdorff G-topology if and only if the Zariski G-topology ζ G is non-discrete; moreover, in this case, X admits 2 c hereditarily normal G-topologies.

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The number of non-solutions of an equation in a group

Cited by 15 publications

References 1 publication

The Markov–Zariski topology of an abelian group

The Markov–Zariski topology of an abelian group

Reflection principle characterizing groups in which unconditionally closed sets are algebraic

Topologizations of a set endowed with an action of a monoid

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