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.

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 Notes on Frattini Subgroups of Generalized Free Products with Cyclic Amalgamation

1980 ◽  
Vol 23 (1) ◽  
pp. 51-59
Author(s):  
R. B. J. T. Allenby ◽  
C. Y. Tang ◽  
S. Y. Tang
Keyword(s):  
Free Product ◽  
Frattini Subgroup ◽  
Free Products ◽  
Exact Location ◽  
Generalized Free Product ◽  
Special Cases ◽  
Generalized Free Products

The problem of the exact location of the Frattini subgroup 4>(G) of a generalized free product G = (A*B)H was first raised by Higman and Neumann [5]. Solutions to special cases of the problem can be found in [1], [2], [8], [9] and [10]. The purpose of this note is to extend the results of [2], [8], and to simplify the proof of Whittemore's theorem [10]. We also apply our result to give simple proofs of certain classes of knot groups that have trivial Frattini subgroups. The proof that every knot group has trivial Frattini subgroup hard and long (footnote 2, p. 56).

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.

scholarly journals Finite and infinite cyclic extensions of free groups

1973 ◽  
Vol 16 (4) ◽  
pp. 458-466 ◽  
Author(s):  
A. Karrass ◽  
A. Pietrowski ◽  
D. Solitar
Keyword(s):  
Finite Number ◽  
Free Group ◽  
Finite Groups ◽  
Free Groups ◽  
Finite Extension ◽  
Finitely Generated ◽  
Free Products ◽  
Tree Product ◽  
Image Position ◽  
Generalized Free Products

Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

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 Permutational Products of Groups:Part 2-amalgamated products

1971 ◽  
Vol 12 (1) ◽  
pp. 21-34
Author(s):  
R. J. Gregorac
Keyword(s):  
Free Product ◽  
Group Structure ◽  
Exact Nature ◽  
Free Products ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Amalgamated Products ◽  
Amalgamated Subgroup ◽  
Apparent Similarity ◽  
The Relationship

The standard methods of constructing generalized free products of groups (with a single amalgamated subgroup) and permutational products of groups are to consider groups of permutations on sets. Although there is an apparent similarity between these two constructions, the exact nature of the relationship is not clear. The following addendum to [4] grew out of an attempt to determine this relationship. By noting that the original construction of permutational products (B. H. Neumann [7]) deals with a group of permutations on a group (although the group structure has been previously ignored; see [7], [8]) we here give an extension of the original permutational product-construction which yields both the generalized free product and the permutational products as groups of permutations on groups. A generalized free product is represented as a group of permutations on the ordinary free product of the constituents of the underlying group amalgam and a permutational product is a group of permutations on the direct product of the constituents of the amalgam.

ON THE INTERSECTION OF FINITELY GENERATED SUBGROUPS IN FREE PRODUCTS OF GROUPS

1999 ◽  
Vol 09 (05) ◽  
pp. 521-528 ◽  
Author(s):  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Free Groups ◽  
Product Formula ◽  
Finitely Generated ◽  
Free Products ◽  
Products Of Groups ◽  
Free Subgroups

A subgroup H of a free product [Formula: see text] of groups Gα, α∈ I, is called factor free if for every [Formula: see text] and β ∈ I one has S H S-1∩ Gβ = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote [Formula: see text], where r(K) is the rank of K. It is proven that if H, K are finitely generated factor free subgroups of a free product [Formula: see text] then [Formula: see text]. It is also shown that the inequality [Formula: see text] of Hanna Neumann conjecture on subgroups of free groups does not hold for factor free subgroups of free products.

scholarly journals On the residual solvability of generalized free products of solvable groups

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Delaram Kahrobaei ◽  
Stephen Majewicz
Keyword(s):  
Free Product ◽  
Solvable Groups ◽  
60Th Birthday ◽  
Special Issue ◽  
Free Products ◽  
Generalized Free Product ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Generalized Free Products

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience In this paper, we study the residual solvability of the generalized free product of solvable groups.

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).

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