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.

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.

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.

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.

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 subgroups of amalgamated free products

1971 ◽  
Vol 69 (1) ◽  
pp. 13-23 ◽  
Author(s):  
Edward T. Ordman
Keyword(s):  
Free Product ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Free Product Of Groups ◽  
Product Of Groups

In 1934 Kurosh(9) proved that ‘a subgroup of a free product of groups is again a free product’. Many other proofs of this, and attempts to generalize it to amalgamated free products, have appeared (e.g. (7), (1), (10) and (8)). Recently the theory of groupoids has been applied to this area with increasing success. In 1966 Higgins (6) used groupoids to prove the generalization of Grushko's Theorem (3) due to Wagner (14).

scholarly journals Free products of inverse semigroups II

1991 ◽  
Vol 33 (3) ◽  
pp. 373-387 ◽  
Author(s):  
Peter R. Jones ◽  
Stuart W. Margolis ◽  
John Meakin ◽  
Joseph B. Stephen
Keyword(s):  
Free Product ◽  
General Result ◽  
Commutative Diagram ◽  
Free Semigroup ◽  
Inverse Semigroups ◽  
Canonical Forms ◽  
Free Products ◽  
Free Product Of Groups ◽  
Product Of Groups

Let S and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product S sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] of Jones and previous partial results [3], [5], [6] take this approach.

Free products and residual nilpotency

1969 ◽  
Vol 1 (1) ◽  
pp. 11-13 ◽  
Author(s):  
I. M. S. Dey
Keyword(s):  
Free Product ◽  
Lower Central Series ◽  
Central Series ◽  
Free Products ◽  
Residual Nilpotency ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups Gα, [Gα] the cartesian subgroup of G and k [Gα] the intersection of [G] with the k–th term of the lower central series for G. Then the k[Gα] form a descending chain of subgroups of ] and it is shown that if the intersection of all the subgroups in this chain is trivial then G and hence each Gα, is residually nilpotent. This answers a question of S. Moran.

scholarly journals Automatic Continuity for Homomorphisms into Free Products

2013 ◽  
Vol 78 (4) ◽  
pp. 1288-1306 ◽  
Author(s):  
Konstantin Slutsky
Keyword(s):  
Free Product ◽  
Discrete Topology ◽  
Group Topology ◽  
Automatic Continuity ◽  
Free Products ◽  
Metrizable Group ◽  
Completely Metrizable ◽  
Free Product Of Groups ◽  
Metrizable Topological Group ◽  
Product Of Groups

AbstractA homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

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