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 the cyclic subgroup separability of the free product of two groups with commuting subgroups

2014 ◽  
Vol 24 (05) ◽  
pp. 741-756 ◽  
Author(s):  
E. V. Sokolov
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Cyclic Subgroups ◽  
Subgroup Separability ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups A and B with commuting subgroups H ≤ A and K ≤ B, and let 𝒞 be the class of all finite groups or the class of all finite p-groups. We derive the description of all 𝒞-separable cyclic subgroups of G provided this group is residually a 𝒞-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p′-isolated in the free factors, then all p′-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p′-isolated and cyclic.

scholarly journals The palindromic width of a free product of groups

2006 ◽  
Vol 81 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Valery Bardakov ◽  
Vladimir Tolstykh
Keyword(s):  
Free Product ◽  
Free Groups ◽  
Uniform Bound ◽  
Free Products ◽  
Cyclic Groups ◽  
Similar Fact ◽  
Products Of Groups ◽  
Free Product Of Groups ◽  
Verbal Subgroups ◽  
Product Of Groups

AbstractPalindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.

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 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.

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.

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.

scholarly journals Faithful, irreducible *-representations for group algebras of free products

1999 ◽  
Vol 42 (3) ◽  
pp. 559-574 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Irreducible Representation ◽  
Free Product ◽  
Irreducible Representations ◽  
Group Algebras ◽  
Free Products ◽  
Characteristic 2 ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups A and B, where |A|≥3 and |B|≥2. We construct faithful, irreducible *-representations for the group algebras ℂ[G] and ℓ1(G). The construction gives a faithful, irreducible representation for F[G] when the field F does not have characteristic 2.

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.

Abstract length functions in groups

1976 ◽  
Vol 80 (3) ◽  
pp. 451-463 ◽  
Author(s):  
I. M. Chiswell
Keyword(s):  
Free Product ◽  
Free Group ◽  
Free Groups ◽  
Arbitrary Group ◽  
Free Products ◽  
Length Functions ◽  
Normal Word ◽  
Free Product Of Groups ◽  
Fixed Basis ◽  
Product Of Groups

If F is a free group on some fixed basis X, there is a mapping from F to the non-negative integers, given by sending an element of F to the length of the normal word in X±1 representing it. A similar mapping is obtained in the case of a free product of groups. Lyndon (3) considered mappings from an arbitrary group to the non-negative integers having certain properties in common with these mappings on free groups and free products.

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.

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