Cyclic Subgroup Separability of Generalized Free Products

Kim, Goansu

doi:10.4153/cmb-1993-042-7

Cited by 14 publications

(9 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, polygonal products of more than four finitely generated abelian groups, amalgamating any subgroups with trivial intersections, are TT C . It was relatively easy to prove the same result for those polygonal products with four vertex groups and cyclic subgroups amalgamated [8]. Note that polygonal products of four polycyclic-by-finite groups amalgamating cyclic subgroups, contained in the centres of their vertex groups, with trivial intersections is conjugacy separable [7].…”

Section: Introductionmentioning

confidence: 73%

“…It was relatively easy to prove the same result for those polygonal products with four vertex groups and cyclic subgroups amalgamated [8]. Note that polygonal products of four polycyclic-by-finite groups amalgamating cyclic subgroups, contained in the centres of their vertex groups, with trivial intersections is conjugacy separable [7]. Unlike the case for residual finiteness or for conjugacy separability, most 454 G. Kim [2] polygonal products of four finitely generated abelian groups amalgamating cyclic subgroups with trivial intersections are not subgroup separable (Theorem 3.3).…”

Section: Introductionmentioning

confidence: 76%

See 1 more Smart Citation

On polygonal products of finitely generated abelian groups

Kim¹

1992

Bull. Austral. Math. Soc.

Self Cite

View full text Add to dashboard Cite

We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

show abstract

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 76%

On polygonal products of finitely generated abelian groups

Kim¹

1992

Bull. Austral. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For each u ∈ U let Z u be a finite generating set for I u , and let Z = u∈U Z u . [17] On separability finiteness conditions in semigroups As U is finite, we have that Z is finite. For each z ∈ Z, we have…”

Section: Corollary 45 Let S Be a Finitely Generated Commutative Semigroup And Let H Be A Nonminimal H-class Let I = I(h) If A Is An Archimentioning

confidence: 99%

On Separability Finiteness Conditions in Semigroups

Miller¹,

O'Reilly²,

Quick³

et al. 2021

J. Aust. Math. Soc.

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 $\mathcal {H}$ -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.

show abstract

“…There certainly has to be some fairly strong condition on the Ki because of the following result in In proving the theorem we shall use the following result of Kim [8].…”

Section: Introductionmentioning

confidence: 99%