scholarly journals Groups whose three-generator subgroups are free

1989 ◽  
Vol 40 (2) ◽  
pp. 163-174 ◽  
Author(s):  
Gilbert Baumslag ◽  
Peter B. Shalen
Keyword(s):  
Free Groups ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Products Of Groups

We define a certain class of groups, Ck, which we show to contain the class of all k-free groups. Our main theorem shows that certain amalgamated free products of groups in C3, are again in C3. In the appendix we show that many 3-manifold groups belong to Ck for suitable k.

Second Order Dehn Functions for Amalgamated Free Products of Groups

2009 ◽  
Vol 16 (04) ◽  
pp. 699-708
Author(s):  
Xiaofeng Wang ◽  
Xiaomin Bao
Keyword(s):  
Free Product ◽  
Upper Bound ◽  
Second Order ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Products Of Groups ◽  
Amalgamated Free Product ◽  
Finite Set

A finite set of generators for a free product of two groups of type F3with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

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.

Units in Group Rings of Free Products of Prime Cyclic Groups

1998 ◽  
Vol 50 (2) ◽  
pp. 312-322 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Maria Lucia Sobral Singer
Keyword(s):  
Unit Group ◽  
Group Rings ◽  
Rational Group ◽  
Free Products ◽  
Structural Result ◽  
Cyclic Groups ◽  
Amalgamated Free Products ◽  
Products Of Groups ◽  
Finite Subgroups ◽  
Zassenhaus Conjecture

AbstractLet G be a free product of cyclic groups of prime order. The structure of the unit group U(ℚG) of the rational group ring ℚG is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of U(ℚG), up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in ℤ G is proved. A strong version of the Tits Alternative for U(ℚG) is obtained as a corollary of the structural result.

Hierarchies of Computable groups and the word problem

1966 ◽  
Vol 31 (3) ◽  
pp. 376-392 ◽  
Author(s):  
Frank B. Cannonito
Keyword(s):  
Word Problem ◽  
Free Groups ◽  
Orientable Surface ◽  
Fundamental Groups ◽  
Church’S Thesis ◽  
Free Products ◽  
Closed Orientable Surface ◽  
Products Of Groups ◽  
Solvable Word Problem ◽  
Solvable Word

The word problem for groups was first formulated by M. Dehn [1], who gave a solution for the fundamental groups of a closed orientable surface of genus g ≧ 2. In the following years solutions were given, for example, for groups with one defining relator [2], free groups, free products of groups with a solvable word problem and, in certain cases, free products of groups with amalgamated subgroups [3], [4], [5]. During the period 1953–1957, it was shown independently by Novikov and Boone that the word problem for groups is recursively undecidable [6], [7]; granting Church's Thesis [8], their work implies that the word problem for groups is effectively undecidable.

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.

Group Properties Characterized by Two-sided Configurations

2010 ◽  
Vol 17 (04) ◽  
pp. 583-594 ◽  
Author(s):  
A. Rejali ◽  
A. Yousofzadeh
Keyword(s):  
Free Groups ◽  
Nilpotent Groups ◽  
Free Products ◽  
Products Of Groups ◽  
Finite Quotient ◽  
New Type ◽  
Torsion Free

In this paper, we define a new type of configurations as two-sided configurations, and investigate which group properties can be characterized by them. It is proved that for polycyclic torsion free groups, having the same finite quotient sets does not imply the (two-sided) configuration equivalence. We show that isomorphisms and configuration equivalences coincide for some free products of groups and a class of nilpotent groups.

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