A RECURRENCE RELATION FOR THE NUMBER OF FREE SUBGROUPS IN FREE PRODUCTS OF CYCLIC GROUPS

2008 ◽  
Author(s):  
T. CAMPS ◽  
M. DÖRFER ◽  
G. ROSENBERGER
Keyword(s):  
Recurrence Relation ◽  
Free Products ◽  
Cyclic Groups ◽  
Free Subgroups

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.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals Stable commutator length in amalgamated free products

2015 ◽  
Vol 07 (04) ◽  
pp. 693-717 ◽  
Author(s):  
Tim Susse
Keyword(s):  
Geometric Model ◽  
Free Products ◽  
Cyclic Groups ◽  
Amalgamated Free Products ◽  
Torus Knot ◽  
Knot Complements ◽  
Stable Commutator Length ◽  
Free Abelian Groups ◽  
Commutator Length ◽  
Boundary Mapping

We show that stable commutator length is rational on free products of free abelian groups amalgamated over ℤk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parametrize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed word varies quasirationally in the orders of the free factors.

Test Elements, Generic Elements and Almost Primitivity in Free Products

2016 ◽  
Vol 23 (02) ◽  
pp. 263-280
Author(s):  
Ann-Kristin Engel ◽  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Free Groups ◽  
Free Products ◽  
Cyclic Groups ◽  
Test Elements ◽  
Generic Elements

In [5, 6] the relationships between test words, generic elements, almost primitivity and tame almost primitivity were examined in free groups. In this paper we extend the concepts and connections to general free products and in particular to free products of cyclic groups.

A note on commutators in free products

1954 ◽  
Vol 50 (2) ◽  
pp. 178-188 ◽  
Author(s):  
H. B. Griffiths
Keyword(s):  
Topological Group ◽  
Free Product ◽  
Commutator Subgroup ◽  
Infinite Sequence ◽  
Natural Topology ◽  
Free Products ◽  
General Sequence ◽  
Cyclic Groups ◽  
The Continuum

In (1), Higman introduces the unrestricted free product of a set of groups, and gives it a natural topology. When this set is an infinite sequence of free cyclic groups he denotes by F their unrestricted free product; we shall denote by K the product for a general sequence {Gn} and, throughout the paper, we assume that each Gn is non-trivial and countable. Higman proves as an incidental result that the commutator subgroup [F, F] is not closed in the topological group F, and the first object of this note is to generalize this result to K. From this, the more interesting deduction immediately follows that K is never equal to L = [K, K]. Indeed, we prove in fact that the cardinal of L and its index in K are both c, the cardinal of the continuum.

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.

Sign in / Sign up

Export Citation Format

Share Document