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.

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.

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.

Generalized partition functions and subgroup growth of free products of nilpotent groups

Analysis ◽  
2005 ◽  
Vol 25 (4) ◽  
Author(s):  
Thomas W. Müller ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Partition Functions ◽  
Free Products ◽  
Subgroup Growth ◽  
Free Nilpotent Group ◽  
Generalized Partition ◽  
Torsion Free

SummaryWe estimate the growth of homomorphism numbers of a torsion-free nilpotent group

Existentially complete torsion-free nilpotent groups

1978 ◽  
Vol 43 (1) ◽  
pp. 126-134 ◽  
Author(s):  
D. Saracino
Keyword(s):  
Free Groups ◽  
Nilpotent Groups ◽  
Generic Model ◽  
Model Companion ◽  
Generic Models ◽  
Direct Sum ◽  
Carry Over ◽  
Torsion Free

This paper continues the study of existentially complete nilpotent groups initiated in [6]. Following [6], we let Kn denote the theory of groups nilpotent of class ≤ n and let Kn+ denote the theory of torsion-free groups nilpotent of class ≤ n. The principal results of [6] were that for n ≥ 2, neither Kn nor Kn+ has a model companion, and the classes E, F, and G of existentially complete, finitely generic and infinitely generic models of Kn are all distinct. The question of the relationships between these classes in the context of Kn was left open, however, and the proof of their distinctness for Kn+ obviously did not carry over to Kn+, because it made strong use of torsion elements.In this paper we establish the relationships between E, F, and G for K2+. We show that all three classes are distinct. We also show that there is only one countable finitely generic model, and only one countable infinitely generic model, and that all the countable existentially complete models can be arranged in a sequence N1 ⊆ N2 ⊆ N3 ⊆ … ⊆ Nω, where Z(Nn) is the direct sum of n copies of Q. Another result is that the finite and infinite forcing companions of K2+ differ by an ∀∃∀ sentence. Finally, we show that there exist finitely generic models of K2+ in all infinite cardinalities.

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 Residual Finiteness of Certain Polygonal Products

1989 ◽  
Vol 32 (1) ◽  
pp. 11-17 ◽  
Author(s):  
R. B. J. T. Allenby ◽  
C. Y. Tang
Keyword(s):  
Abelian Groups ◽  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Residually Finite ◽  
Products Of Groups ◽  
Generalized Free Products

AbstractWe give examples to show that unlike generalized free products of groups (g.f.p.) polygonal products of finitely generated (f.g.) nilpotent groups with cyclic amalgamations need not be residually finite (R) and polygonal products of finite p-groups with cyclic amalgamations need not be residually nilpotent. However, polygonal products f.g. abelian groups are R, and under certain conditions polygonal products of finite p-groups with cyclic amalgamations are R.

scholarly journals Recognizing powers in nilpotent groups and nilpotent images of free groups

2007 ◽  
Vol 83 (2) ◽  
pp. 149-156
Author(s):  
Gilbert Baumslag
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Proper Power ◽  
Torsion Free

AbstractAn element in a free group is a proper power if and only if it is a proper power in every nilpotent factor group. Moreover there is an algorithm to decide if an element in a finitely generated torsion-free nilpotent group is a proper power.

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.

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