Residual finiteness, subgroup separability and conjugacy separability of certain HNN extensions

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
K. Wong ◽  
P. Wong
Keyword(s):  
Finite Groups ◽  
Finite Subgroup ◽  
Residual Finiteness ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Conjugacy Separability ◽  
Subgroup Separability ◽  
Conjugacy Separable

AbstractIn this note we shall give characterisations for HNN extensions of non-cyclic polycyclic-by-finite groups with normal infinite cyclic associated subgroups to be residually finite, subgroup separable and conjugacy separable.

Conjugacy Separability of Certain Polygonal Products

1996 ◽  
Vol 39 (3) ◽  
pp. 294-307 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Cyclic Subgroups ◽  
Conjugacy Separability ◽  
Conjugacy Separable

AbstractWe show that polygonal products of polycyclic-by-finite groups amalgamating central cyclic subgroups, with trivial intersections, are conjugacy separable. Thus polygonal products of finitely generated abelian groups amalgamating cyclic subgroups, with trivial intersections, are conjugacy separable. As a corollary of this, we obtain that the group A1 *〈a1〉A2 *〈a2〉 • • • *〈am-1〉Am is conjugacy separable for the abelian groups Ai.

Residually finite groups of finite rank

1989 ◽  
Vol 106 (3) ◽  
pp. 385-388 ◽  
Author(s):  
Alexander Lubotzky ◽  
Avinoam Mann
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Rank ◽  
Prime Order ◽  
Residual Finiteness ◽  
Infinite Groups ◽  
Finiteness Conditions ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:Theorem 1. A residually finite group of finite rank is virtually locally soluble.

Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups

1999 ◽  
Vol 42 (3) ◽  
pp. 335-343 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Residual Finiteness ◽  
Necessary And Sufficient Condition ◽  
Residual Properties ◽  
Hnn Extensions ◽  
Subgroup Separability ◽  
Necessary And Sufficient

AbstractWe derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties ofHNN-extensions of nilpotent groups with cyclic associated subgroups.

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.

ON RESIDUAL FINITENESS OF GRAPHS OF NILPOTENT GROUPS

2004 ◽  
Vol 14 (04) ◽  
pp. 403-408
Author(s):  
E. RAPTIS ◽  
O. TALELLI ◽  
D. VARSOS
Keyword(s):  
Finite Groups ◽  
Finite Graph ◽  
Nilpotent Groups ◽  
Fundamental Groups ◽  
Residual Finiteness ◽  
Graph Of Groups ◽  
Residually Finite ◽  
Edge Group ◽  
Residually Finite Groups ◽  
Torsion Free

Here we characterize the residually finite groups G which are the fundamental groups of a finite graph of finitely generated torsion-free nilpotent groups. Namely we show that G is residually finite if and only if for each edge group of the graph of groups the two edge monomorphisms differ essentially by an isomorphism of certain subgroups of the Mal'cev completion of the corresponding vertex groups.

RESIDUAL FINITENESS OF OUTER AUTOMORPHISM GROUPS OF FINITELY GENERATED NON-TRIANGLE FUCHSIAN GROUPS

2005 ◽  
Vol 15 (01) ◽  
pp. 59-72 ◽  
Author(s):  
R. B. J. T. ALLENBY ◽  
GOANSU KIM ◽  
C. Y. TANG
Keyword(s):  
Automorphism Groups ◽  
Outer Automorphism ◽  
Fuchsian Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Outer Automorphism Groups ◽  
Orientable Surfaces

In [5] Grossman showed that outer automorphism groups of free groups and of fundamental groups of compact orientable surfaces are residually finite. In this paper we introduce the concept of "Property E" of groups and show that certain generalized free products and HNN extensions have this property. We deduce that the outer automorphism groups of finitely generated non-triangle Fuchsian groups are residually finite.

scholarly journals Residual finiteness of permutational products

1969 ◽  
Vol 10 (3-4) ◽  
pp. 423-428 ◽  
Author(s):  
R. J. Gregorac
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Residual Finiteness ◽  
Residually Finite ◽  
Generalized Free Product ◽  
Residually Finite Groups

Conditions sufficient to guarantee that a generalized free product of two residually finite groups A and B is again residually finite have been given by Baumslag [1]. We here show the same conditions guarantee that a certain permutational product of A and B is also residually finite.

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