Residual finiteness of the free product of two groups with commuting subgroups

Loginova, E. D.

doi:10.1007/s11202-999-0013-8

Cited by 22 publications

(14 citation statements)

References 5 publications

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“…Baumslag [10] studied in one special class of one-relator groups. Loginova obtained in [20] a similar result for a larger class of groups. The paper [24] gives a characterization of residually nilpotent one-relator groups with non-trivial center.…”

Section: Introductionsupporting

confidence: 60%

On residually nilpotence of group extensions

Bardakov,

Neshchadim,

Bryukhanov

2021

Preprint

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We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: F n ⋊ ϕ Z. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for n = 2 the length of the lower central series of F n ⋊ ϕ Z is equal to 2, ω or ω 2 .

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Section: Introductionsupporting

confidence: 60%

On residually nilpotence of group extensions

Bardakov,

Neshchadim,

Bryukhanov

2021

Preprint

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“…Let also Ω p A and Ω p B denote the families of the first and the second components of elements of Ω p . The next proposition is obtained by E. D. Loginova in the paper [4].…”

Section: Introduction Main Resultsmentioning

confidence: 79%

On the cyclic subgroup separability of free products of two groups with amalgamated subgroup

Sokolov

2007

Preprint

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Let G be a free product of two groups with amalgamated subgroup, π be either the set of all prime numbers or the one-element set {p} for some prime number p. Denote by Σ the family of all cyclic subgroups of group G, which are separable in the class of all finite π-groups.Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite π-index of group G, the subgroups conjugated with them and all subgroups, which aren't π ′ -isolated, don't belong to Σ. Some sufficient conditions are obtained for Σ to coincide with the family of all other π ′ -isolated cyclic subgroups of group G.It is proved, in particular, that the residual p-finiteness of a free product with cyclic amalgamation implies the p-separability of all p ′ -isolated cyclic subgroups if the free factors are free or finitely generated residually p-finite nilpotent groups.1991 Mathematics Subject Classification. 20E06, 20E26 (primary).

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“…Recall that the length of an element of T is the length of any reduced form of this element. It is easy to see [5] that the structure of G can be described in terms of the construction of free product with amalgamated subgroup as follows.…”

Section: The Main Theoremmentioning

confidence: 99%

“…It is known [5] that all π ′ -isolated subgroups of a finitely generated nilpotent group are F π -separable. So there exists a subgroup L ∈ Ω π ( u, k ) such that g / ∈ CL.…”

Section: Some Corollariesmentioning

confidence: 99%

On the cyclic subgroup separability of the free product of two groups with commuting subgroups

Sokolov

2014

Int. J. Algebra Comput.

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Let G be the free product of groups A and B with commuting subgroups H A and K B, and let C be the class of all finite groups or the class of all finite p-groups. We derive the description of all C-separable cyclic subgroups of G provided this group is residually a C-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p ′ -isolated in the free factors, then all p ′ -isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p ′ -isolated and cyclic.

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Residual finiteness of the free product of two groups with commuting subgroups

Cited by 22 publications

References 5 publications

On residually nilpotence of group extensions

On residually nilpotence of group extensions

On the cyclic subgroup separability of free products of two groups with amalgamated subgroup

On the cyclic subgroup separability of the free product of two groups with commuting subgroups

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