The Root Class Residuality of the Tree Product of Groups with Amalgamated Retracts

Tumanova, E. A.

doi:10.1134/s0037446619040153

Cited by 14 publications

(5 citation statements)

References 21 publications

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“…The use of the concept of a root class turned out to be very productive in the study of some residual properties of free constructions of groups: generalized free and tree products, HNN-extensions, fundamental groups of graphs of groups, etc. It allows one to prove several statements at once and quickly complicate the constructions under consideration; see, for example, [40,[42][43][44][45][46][47][48]. At the same time, if C is a root class different from the class of all finite groups, very few facts are known about the conjugacy C-separability of free constructions of groups.…”

Section: Introduction Statement Of Resultsmentioning

confidence: 99%

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Certain residual properties of generalized Baumslag–Solitar groups

Sokolov

2021

Journal of Algebra

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Section: Introduction Statement Of Resultsmentioning

confidence: 99%

“…[48, Proposition 8] Let C be a root class of groups consisting only of periodic groups. A finite solvable group belongs to C if and only if it is a P(C)-group.If m, n ∈ Z and 0 < m |n|, then the following statements hold.…”

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confidence: 99%

Certain residual properties of generalized Baumslag–Solitar groups

Sokolov

2021

Journal of Algebra

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“…The notion of a root class was introduced in [10] and allows one to prove many statements at once using the same reasoning. It turns out to be especially useful in studying the residual properties of free constructions of groups (see, e.g., [22][23][24][25][26][27][28][29][30][31]). If X is such a construction and C is a root class of groups, then the C-separability of some subgroups of X is quite often one of the necessary and/or sufficient conditions for X to be residually a C-group.…”

Section: Introductionmentioning

confidence: 99%

On the separability of subgroups of nilpotent groups by root classes of groups

Sokolov¹

2022

Preprint

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Suppose that C is a class of groups consisting only of periodic groups and P(C) ′ is the set of prime numbers each of which does not divide the order of any element of a C-group. It is easy to see that if a subgroup Y of a group X is C-separable in this group, then it is P(C) ′ -isolated in X. Let us say that X has the property C-Sep if all its P(C) ′ -isolated subgroups are C-separable. We find a condition that is sufficient for a nilpotent group N to have the property C-Sep provided C is a root class. We also prove that if N is torsion-free, then the indicated condition is necessary for this group to have C-Sep.

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“…We strengthen some known results (for example, on the residual finiteness results at once using the same reasoning. This approach was originally proposed in [12,21] and turned out to be very fruitful in the study of free constructions of groups in the case when C was a root class; see, e. g. [2,[25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Certain residual properties of generalized Baumslag-Solitar groups

Sokolov

2020

Preprint

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Let G be a generalized Baumslag-Solitar group and C be a class of groups containing at least one non-unit group and closed under taking subgroups, extensions, and Cartesian products of the form y∈Y X y , where X, Y ∈ C and X y is an isomorphic copy of X for every y ∈ Y . We give a criterion for G to be residually a C-group provided C consists only of periodic groups. We also prove that G is residually a torsion-free C-group if C contains at least one non-periodic group and is closed under taking homomorphic images. These statements generalize and strengthen some known results. Using the first of them, we provide criteria for a GBS-group to be a) residually nilpotent; b) residually torsion-free nilpotent; c) residually free. and the residual p-finiteness of GBS-groups) and give a criterion for the residual nilpotence of a GBS-group.Let C be a class of groups. A group G is said to be residually a C-group if, for any non-unit element g ∈ G, there exists a homomorphism σ of G onto a group of C such that gσ = 1. The most commonly considered situation is when C is the class of all finite groups, all finite p-groups (where p is a prime number), all nilpotent groups or all solvable groups. In these cases G is called residually finite, residually p-finite, residually nilpotent or residually solvable respectively.We say that a class C of groups is root if it contains at least one non-unit group and is closed under taking subgroups, extensions, and Cartesian products of the form y∈Y X y , where X, Y ∈ C and X y is an isomorphic copy of X for every y ∈ Y . The notion of a root class was introduced by Gruenberg [12], and the above definition is equivalent to that given in [12]; see [25] for details.The classes of all finite groups, all finite p-groups, all periodic groups of finite exponent, all solvable groups, and all torsion-free groups can serve as examples of root classes. It is also easy to see that the intersection of any number of root classes is again a root class. At the same time, the classes of all nilpotent groups, all torsion-free nilpotent groups, and all finite nilpotent groups are not root because they are not closed under taking extensions.

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The Root Class Residuality of the Tree Product of Groups with Amalgamated Retracts

Cited by 14 publications

References 21 publications

Certain residual properties of generalized Baumslag–Solitar groups

Certain residual properties of generalized Baumslag–Solitar groups

On the separability of subgroups of nilpotent groups by root classes of groups

Certain residual properties of generalized Baumslag-Solitar groups

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