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.

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.

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.

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.

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.

PROFINITE TOPOLOGIES IN FREE PRODUCTS OF GROUPS

2004 ◽  
Vol 14 (05n06) ◽  
pp. 751-772 ◽  
Author(s):  
LUIS RIBES ◽  
PAVEL ZALESSKII
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Free Products ◽  
Products Of Groups

Let [Formula: see text] be a nonempty class of finite groups closed under taking subgroups, quotients and extensions. We consider groups G endowed with their pro-[Formula: see text] topology, and say that G is 2-subgroup separable if whenever H and K are finitely generated closed subgroups of G, then the subset HK is closed. We prove that if the groups G1 and G2 are 2-subgroup separable, then so is their free product G1*G2. This extends a result to T. Coulbois. The proof uses actions of groups on abstract and profinite trees.

Sign in / Sign up

Export Citation Format

Share Document