Subgroup Separability of Certain HNN Extensions

Wong, P. C.

doi:10.1216/rmjm/1181072631

Cited by 9 publications

(4 citation statements)

References 8 publications

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“…All known answers to this question are obtained under various restrictions imposed on the base group, the associated subgroups, and / or the isomorphism between them (the terminology for HNN-extensions used here and below follows [9]). One such restriction is the centrality of the associated subgroups in the base group; the C-residuality of HNN-extensions of this type (including the case of an abelian base group) is studied in [2,6,8,[11][12][13][14][15]23,24,33,34].…”

Section: Introduction Statement Of Resultsmentioning

confidence: 99%

Root class residuality of HNN-extensions with central cyclic associated subgroups

Sokolov

Tumanova

2017

Math Notes

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Suppose that G is a group, H and K are proper isomorphic central subgroups of G, and G is an HNN-extension of G with the associated subgroups H and K. We prove necessary and sufficient conditions for G to be residually a C-group, where C is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form y∈Y X y , where X, Y ∈ C and X y is an isomorphic copy of X for each y ∈ Y .

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Section: Introduction Statement Of Resultsmentioning

confidence: 99%

Root class residuality of HNN-extensions with central cyclic associated subgroups

Sokolov

Tumanova

2017

Math Notes

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“…Groups satisfying the separability property with respect to the collection of all cyclic subgroups are known in the literature as cyclic subgroup separable groups or groups . Such groups have received considerable attention: see, for instance, [2, 17, 28, 29].…”

Section: Schützenberger Groupsmentioning

confidence: 99%

On Separability Finiteness Conditions in Semigroups

Miller¹,

O'Reilly²,

Quick³

et al. 2021

J. Aust. Math. Soc.

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Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every $\mathcal {H}$ -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal {H}$ -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

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“…Groups satisfying the separability property with respect to the collection of all cyclic subgroups are known in the literature as cyclic subgroup separable groups or Π c groups. Such groups have received considerable attention; see for instance [2,18,29,30].…”

Section: Schützenberger Groupsmentioning

confidence: 99%

On separability finiteness conditions in semigroups

Miller¹,

O’Reilly²,

Quick³

et al. 2020

Preprint

View full text Add to dashboard Cite

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H-class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H-classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

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Subgroup Separability of Certain HNN Extensions

Cited by 9 publications

References 8 publications

Root class residuality of HNN-extensions with central cyclic associated subgroups

Root class residuality of HNN-extensions with central cyclic associated subgroups

On Separability Finiteness Conditions in Semigroups

On separability finiteness conditions in semigroups

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