On the Residual Finiteness of Certain Polygonal Products

1989 ◽  
Vol 32 (1) ◽  
pp. 11-17 ◽  
Author(s):  
R. B. J. T. Allenby ◽  
C. Y. Tang
Keyword(s):  
Abelian Groups ◽  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Residually Finite ◽  
Products Of Groups ◽  
Generalized Free Products

AbstractWe give examples to show that unlike generalized free products of groups (g.f.p.) polygonal products of finitely generated (f.g.) nilpotent groups with cyclic amalgamations need not be residually finite (R) and polygonal products of finite p-groups with cyclic amalgamations need not be residually nilpotent. However, polygonal products f.g. abelian groups are R, and under certain conditions polygonal products of finite p-groups with cyclic amalgamations are R.

scholarly journals The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

1992 ◽  
Vol 53 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Raptis ◽  
D. Varsos
Keyword(s):  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Hnn Extensions ◽  
Base Group ◽  
Generalized Free Products

AbstractWe study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

The Residual Finiteness of Polygonal Products—Two Counterexamples

1994 ◽  
Vol 37 (4) ◽  
pp. 433-436 ◽  
Author(s):  
R. B. J. T. Allenby
Keyword(s):  
Abelian Groups ◽  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe show that, even under very favourable hypotheses, a polygonal product of finitely generated torsion free nilpotent groups amalgamating infinite cyclic subgroups is, in general, not residually finite, thus answering negatively a question of C. Y. Tang. A second example shows similar kinds of limitations apply even when the factors of the product are free abelian groups.

On the Residual Finiteness of Polygonal Products of Nilpotent Groups

1992 ◽  
Vol 35 (3) ◽  
pp. 390-399 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Nilpotent Groups ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Isolated Subgroup ◽  
Torsion Free

AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

scholarly journals Near Frattini subgroups of residually finite generalized free products of groups

2001 ◽  
Vol 26 (2) ◽  
pp. 117-121
Author(s):  
Mohammad K. Azarian
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Frattini Subgroup ◽  
Free Products ◽  
Residually Finite ◽  
Identical Relation ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Amalgamated Subgroup ◽  
Generalized Free Products

LetG=A★HBbe the generalized free product of the groupsAandBwith the amalgamated subgroupH. Also, letλ(G)andψ(G)represent the lower near Frattini subgroup and the near Frattini subgroup ofG, respectively. IfGis finitely generated and residually finite, then we show thatψ(G)≤H, providedHsatisfies a nontrivial identical relation. Also, we prove that ifGis residually finite, thenλ(G)≤H, provided: (i)Hsatisfies a nontrivial identical relation andA,Bpossess proper subgroupsA1,B1of finite index containingH; (ii) neitherAnorBlies in the variety generated byH; (iii)H<A1≤AandH<B1≤B, whereA1andB1each satisfies a nontrivial identical relation; (iv)His nilpotent.

scholarly journals Separability properties of automorphisms of graph products of groups

2016 ◽  
Vol 26 (01) ◽  
pp. 1-27
Author(s):  
Michal Ferov
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Technical Condition ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Residually Finite ◽  
Products Of Groups ◽  
Text Version ◽  
Inner Automorphisms

We study properties of automorphisms of graph products of groups. We show that graph product [Formula: see text] has nontrivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has nontrivial pointwise inner automorphisms. We use this result to study residual finiteness of [Formula: see text]. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then [Formula: see text] is residually finite. Finally, we generalize this result to graph products of residually [Formula: see text]-finite groups to show that if [Formula: see text] is a graph product of finitely generated residually [Formula: see text]-finite groups such that the vertex groups corresponding to central vertices satisfy the [Formula: see text]-version of the technical condition then [Formula: see text] is virtually residually [Formula: see text]-finite. We use this result to prove bi-orderability of Torreli groups of some graph products of finitely generated residually torsion-free nilpotent 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.

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