scholarly journals The Cyclic Subgroup Separability of Certain Generalized Free Products of Two Groups

2010 ◽  
Vol 17 (04) ◽  
pp. 577-582 ◽  
Author(s):  
P. A. Bobrovskii ◽  
E. V. Sokolov
Keyword(s):  
Finite Groups ◽  
Cyclic Subgroup ◽  
Free Products ◽  
Residually Finite ◽  
Residually Finite Groups ◽  
Subgroup Separability ◽  
Generalized Free Products

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.

Subgroup Separability of Generalized Free Products of Free-By-Finite Groups

1993 ◽  
Vol 36 (4) ◽  
pp. 385-389 ◽  
Author(s):  
R. B. J. T. Allenby ◽  
C. Y. Tang
Keyword(s):  
Finite Groups ◽  
Word Problems ◽  
Cyclic Subgroup ◽  
Finitely Generated ◽  
Free Products ◽  
Subgroup Separability ◽  
Solvable Word ◽  
Generalized Free Products

AbstractWe prove that generalized free products of finitely generated free-byfinite groups amalgamating a cyclic subgroup are subgroup separable. From this it follows that if where t ≥ 1 and u, v are words on {a1,...,am} and {b1,...,bn} respectively then G is subgroup separable thus generalizing a result in [9] that such groups have solvable word problems.

Conjugacy Separability of Generalized Free Products of Certain Conjugacy Separable Groups

1995 ◽  
Vol 38 (1) ◽  
pp. 120-127 ◽  
Author(s):  
C. Y. Tang
Keyword(s):  
Finite Groups ◽  
Cyclic Subgroup ◽  
Finitely Generated ◽  
Free Products ◽  
Conjugacy Separability ◽  
Conjugacy Separable ◽  
Generalized Free Products

AbstractWe prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finite groups amalgamating a cyclic subgroup areconjugacy separable. Applying this result we prove a generalization of a conjecture of Fine and Rosenberger [7] that groups of F-type are conjugacy separable.

Abelian Subgroup Separability of Certain Generalized Free Products of Groups

2020 ◽  
Vol 27 (04) ◽  
pp. 651-660
Author(s):  
Wei Zhou ◽  
Goansu Kim
Keyword(s):  
Finite Groups ◽  
Abelian Subgroup ◽  
Free Products ◽  
Products Of Groups ◽  
Subgroup Separability ◽  
Generalized Free Products

We prove that generalized free products of certain abelian subgroup separable groups are abelian subgroup separable. Applying this, we show that tree products of polycyclic-by-finite groups, amalgamating central subgroups or retracts are abelian subgroup separable.

Packings, free products and residually finite groups

1963 ◽  
Vol 59 (3) ◽  
pp. 555-558 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Cardinal Number ◽  
Weak Sense ◽  
Free Products ◽  
Residually Finite ◽  
Residual Properties ◽  
Residually Finite Groups ◽  
The Family ◽  
Discontinuous Groups

In this note a simple principle is explained for constructing a transformation group which is a free product of given transformation groups. The principle does not seem to have been formulated explicitly, though it has been used in a more or less vague form in the theory of discontinuous groups (see, for instance, L. R. Ford, Automorphic functions, vol. I, pp. 56–59). It is perhaps of interest that the formulation given here is purely set-theoretic, without any topology, and that it can apply to any free product, whatever the cardinal number of the set of factors. The principle is used to establish the closure under the formation of countable free product of the family of groups which can be represented as discontinuous subgroups of a certain group of rational projective transformations. (The word ‘discontinuous’ is used here in a weak sense, defined later.) Finally, these results are applied to give a new proof of the theorem of Gruenberg that a free product of residually finite groups is itself residually finite (K. W. Gruenberg: Residual properties of groups, Proc. London Math. Soc. (3), 7 (1957), 29–62. See Corollary (ii) of Theorem 4.1, p. 44). The present proof is completely different from Gruenberg's and seems to be of interest for its own sake, though it does not appear to lead to Gruenberg's other results in this connexion. I have to thank Mr W. J. Harvey and Mr C. Maclachlan for checking a first draft of this paper and pointing out a few errors.

Cyclic Subgroup Separability of Generalized Free Products

1993 ◽  
Vol 36 (3) ◽  
pp. 296-302 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Free Product ◽  
Cyclic Subgroup ◽  
Free Products ◽  
Cyclic Subgroups ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Subgroup Separability ◽  
Free Product Of Groups ◽  
Generalized Free Products ◽  
Product Of Groups

AbstractWe derive a criterion for a generalized free product of groups to be cyclic subgroup separable. We see that most of the known results for cyclic subgroup separability are covered by this criterion, and we apply the criterion to polygonal products of groups. We show that a polygonal product of finitely generated abelian groups, amalgamating cyclic subgroups, is cyclic subgroup separable.

scholarly journals On Groups of Automorphisms of Residually Finite Groups

2000 ◽  
Vol 231 (2) ◽  
pp. 561-573
Author(s):  
Ulderico Dardano ◽  
Bettina Eick ◽  
Martin Menth
Keyword(s):  
Finite Groups ◽  
Residually Finite ◽  
Residually Finite Groups ◽  
Groups Of Automorphisms

scholarly journals An uncountable family of finitely generated residually finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hip Kuen Chong ◽  
Daniel T. Wise
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Uncountable Family ◽  
Residually Finite Groups ◽  
Rank 2

Abstract We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

Sign in / Sign up

Export Citation Format

Share Document