On a problem on explicit embeddings of the group <i>ℚ</i>

Mikaelian, Vahagn H.

doi:10.1155/ijmms.2005.2119

Cited by 11 publications

(12 citation statements)

References 12 publications

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“…Moreover, the group Q coincides with the centre of A Q (m, n). We note that another explicit (subnormal) embedding of the group Q into some finitely generated group was previously proposed in [10]. An effective embedding of the group Q into a finitely generated group can also be found in the paper [11].…”

Section: Introductionmentioning

confidence: 78%

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Central extensions of free periodic groups

Адян¹,

Atabekyan²

2018

Sb. Math.

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It is proved that any countable abelian group D can be embedded as a centre into a m-generated group A such that the quotient group A/D is isomorphic to the free Burnside group B(m, n) of rank m > 1 and of odd period n ≥ 665. The proof is based on some modification of the method which was used by S.I.Adian in his monograph in 1975 for a positive solution of Kontorovich's famous problem from Kourovka Notebook on the existence of a finitely generated non-commutative analogue of the additive group of rational numbers. More precisely, he proved that the desired analogues in which the intersection of any two non-trivial subgroups is infinite, can be constructed as a central extension of the free Burnside group B(m, n), where m > 1, and n ≥ 665 is an odd number, using as its center the infinite cyclic group. The paper also discusses other applications of the proposed generalization of Adian's technique. In particular, we describe the free groups of the variety defined by the identity [x n , y] = 1 and the Schur multipliers of the free Burnside groups B(m, n) for any odd n ≥ 665.

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Section: Introductionmentioning

confidence: 78%

“…Then we assume that Qd P (d ′ d). To prove that after this factorization we obtain a group isomorphic to the group A D (m, n) and that the group D is embedded in it, in particular, we have to prove the relations (10) and (11) together with the concomitant statements.…”

Section: Adian-novikov Theorymentioning

confidence: 99%

Central extensions of free periodic groups

Адян¹,

Atabekyan²

2018

Sb. Math.

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“…), or when the group T has a required property, including those inherited from G (is soluble, is generalized soluble or generalized nilpotent, is linerly ordered, residualy has a property, is simple, etc.). For an outline of the topic see articles [5,12,21,22,23,24,29,14,15,16,20] and the literature cited therein.…”

Section: Introductionmentioning

confidence: 99%

“…We would like to stress the following case related to the question of de la Harpe mentioned as Problem 14.10 (b) in the Kourovka Notebook [13]: "Find an explicit embedding of in a finitely generated group; such a group exists by Theorem IV in [7]". The required explicit embedding of into a 2-generator group T was given in [16] in two manners, using free constructions and wreath products. In the current note we add one more feature: can be explicitly embedded into such a group T the defining relations of which can also be explicitly listed.…”

Section: Introductionmentioning

confidence: 99%

Embeddings using universal words in the free group of rank 2

Mikaelian¹

2020

Preprint

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For a countable group G = 〈 A | R 〉 presented by its generators A and defining relations R we discuss a simple method of embedding for G into such a 2-generator group T that the images of generators from A are explicitly given in T , and the defining relations for T can automatically be deduced from the relations R. The obtained method is applied on particular examples of groups, and references of its application for embeddings of recursive groups into finitely presented groups are given.

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“…later supplied by Mikaelian in 2005 [34], but no solution to part (a) of the problem has appeared in the literature.…”

Section: Introductionmentioning

confidence: 99%

Embedding $\mathbb{Q}$ into a Finitely Presented Group

Belk¹,

Howard²,

Matucci³

2020

Preprint

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We prove that the lift of Thompson's group T to the real line contains the additive group Q of the rational numbers. This gives an explicit, natural example of a finitely presented group that contains Q, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.We also prove that Q can be embedded into a finitely presented simple group. Specifically, we describe a simple group TA of homeomorphisms of the circle that contains Q, and we prove that TA is two-generated and has type F∞. Our method for proving finiteness properties extends existing techniques to allow for groups of homeomorphisms that are locally more complicated than the standard Thompson groups.

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On a problem on explicit embeddings of the group ℚ

Cited by 11 publications

References 12 publications

Central extensions of free periodic groups

Central extensions of free periodic groups

Embeddings using universal words in the free group of rank 2

Embedding $\mathbb{Q}$ into a Finitely Presented Group

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