On the Root-Class Residuality of Certain Free Products of Groups with Normal Amalgamated Subgroups

Sokolov, E. V.; Tumanova, E. A.

doi:10.3103/s1066369x20030044

“…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%

“…[29, Proposition 17] If C is a root class of groups consisting only of periodic groups, then an arbitrary C-group is of finite exponent. Proposition 4.3.…”

mentioning

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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“…At the same time, the approximability by other classes of groups is also considered in the literature, and many of these classes are root classes of groups.In accordance with one of the equivalent definitions (see Proposition 3.2 below), a class of groups C is called a root class if it contains non-trivial groups 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 each y ∈ Y . The concept of a root class was introduced by K. Gruenberg [5] and turned out to be very useful in studying the approximability of the fundamental groups of various graphs of groups [1,4,[13][14][15][16][17][18]21,22]. Thanks to its use, it became possible, in particular, to make significant progress in the study of the residual p-finiteness (where p is a prime number) and the residual solvability of such groups.Everywhere below, it is assumed that Γ = (V, E) is a non-empty connected undirected graph with a vertex set V and an edge set E (loops and multiple edges are allowed).…”

mentioning

confidence: 99%

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

Sokolov

¹

2023

Journal of Group Theory

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Suppose that 𝒞 is a class of groups consisting only of periodic groups and P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} is the set of prime numbers that do 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 ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated in 𝑋. Let us say that 𝑋 has the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} if all of its P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated subgroups are 𝒞-separable. We find a condition that is sufficient for a nilpotent group 𝑁 to have the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{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 ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} .

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On the Root-Class Residuality of Certain Free Products of Groups with Normal Amalgamated Subgroups

Cited by 11 publications

References 17 publications

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

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

Certain residual properties of generalized Baumslag-Solitar groups

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

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