On the Frattini and upper near Frattini subgroups of a generalized free product

2008 ◽  
Vol 11 (3) ◽  
Author(s):  
R. B. J. T. Allenby
Keyword(s):  
Free Product ◽  
Generalized Free Product

scholarly journals On the Root-Class Residuallity of Generalized Free Products

2015 ◽  
Vol 20 (1) ◽  
pp. 133-137 ◽  
Author(s):  
E. A. Tumanova
Keyword(s):  
Free Product ◽  
Sufficient Condition ◽  
Free Products ◽  
Generalized Free Product ◽  
Amalgamated Subgroup ◽  
Root Class ◽  
Generalized Free Products

Let K be a root class of groups. It is proved that a free product of any family of residually K groups with one amalgamated subgroup, which is a retract in all free factors, is residually K. The sufficient condition for a generalized free product of two groups to be residually K is also obtained, provided that the amalgamated subgroup is normal in one of the free factors and is a retract in another.

Permutational products of groups

1960 ◽  
Vol 1 (3) ◽  
pp. 299-310 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Free Product ◽  
Homomorphic Image ◽  
Classical Theorem ◽  
Normal Subgroups ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Amalgamated Subgroup

We deal with questions about the possible embeddings of two given groups A and B in a group P such that the intersection of A and B is a given subgroup H. The data, consisting of the “constituents” A and B with the “amalgamated” subgroup H, form an amalgam.1 According to a classical theorem of Otto Schreier [5], every amalgam of two groups can be embedded in a group F, the “free product of A and B with amalgamated subgroup H” or the “generalized free product” of the amalgam. This has the property that every group P in which the amalgam is embedded and which is generated by the amalgam, is a homomorphic image of it. Hence theorems on the existence of certain embedding groups P can be interpreted also as theorems on the existence of certain normal subgroups of F.

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.

Fuchsian Subgroups of the Picard Group

1976 ◽  
Vol 28 (3) ◽  
pp. 481-485 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Free Product ◽  
Semidirect Product ◽  
Finite Index ◽  
Function Theory ◽  
Picard Group ◽  
Automorphic Function ◽  
Linear Transformations ◽  
Gaussian Integers ◽  
Abstract Group ◽  
Generalized Free Product

The Picard group Γ = PSL2 (Z(i)) is the group of linear transformationswith a, b, c, d Gaussian integers.Γ is of interest both as an abstract group and in automorphic function theory [10]. In [10] Waldinger constructed a subgroup H of finite index which is a generalized free product, while in [1] Fine showed that T is a semidirect product with the subgroup H, contained as a subgroup of finite index in the normal factor.

On the Frattini Subgroups of Generalized Free Products with Cyclic Amalgamations(1)

1972 ◽  
Vol 15 (4) ◽  
pp. 569-573 ◽  
Author(s):  
C. Y. Tang
Keyword(s):  
Free Product ◽  
Free Groups ◽  
Maximal Subgroups ◽  
Finitely Generated ◽  
Frattini Subgroup ◽  
Free Products ◽  
Generalized Free Product ◽  
Special Cases ◽  
Amalgamated Subgroup ◽  
Generalized Free Products

In [1] Higman and Neumann asked the questions whether the Frattini subgroup of a generalized free product can be larger than the amalgamated subgroup and whether such groups necessarily have maximal subgroups. In [4] Whittemore gave answers to the special cases of generalized free products of finitely many free groups with cyclic amalgamation and of generalized free products of finitely many finitely generated abelian groups. In this paper we shall study the Frattini subgroups of generalized free products of any groups with cyclic amalgamation.

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.

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