SOME PROPERTIES OF FREE GROUPS OF SOME SOLUBLE VARIETIES OF GROUPS

2001 ◽  
Vol 63 (3) ◽  
pp. 592-606
Author(s):  
DANIEL GROVES
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Lower Central Series ◽  
Central Series ◽  
Lie Rings ◽  
Torsion Free

Let F be a free group, and let γn(F) be the nth term of the lower central series of F. It is proved that F/[γj(F), γi(F), γk(F)] and F/[γj(F), γi(F), γk(F), γl(F)] are torsion free and residually nilpotent for certain values of i, j, k and i, j, k, l, respectively. In the process of proving this, it is proved that the analogous Lie rings are torsion free.

Massey Products and Lower Central Series of Free Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 322-337 ◽  
Author(s):  
Roger Fenn ◽  
Denis Sjerve
Keyword(s):  
Free Group ◽  
Differential Calculus ◽  
Free Groups ◽  
Lower Central Series ◽  
Central Series ◽  
Massey Product ◽  
Massey Products ◽  
One Dimensional ◽  
Image Position

The purpose of this paper is to continue the investigation into the relationships amongst Massey products, lower central series of free groups and the free differential calculus (see [4], [9], [12]). In particular we set forth the notion of a universal Massey product ≪α1, …, αk≫, where the αi are one dimensional cohomology classes. This product is defined with zero indeterminacy, natural and multilinear in its variables.In order to state the results we need some notation. Throughout F will denote the free group on fixed generators x1, …, xn andwill denote the lower central series of F. If I = (i1, …, ik) is a sequence such that 1 ≦ i1, …, ik ≦ n then ∂1 is the iterated Fox derivative and , where is the augmentation. By convention we set ∂1 = identity if I is empty.

scholarly journals Filtrations of free groups arising from the lower central series

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Michael Chapman ◽  
Ido Efrat
Keyword(s):  
Free Group ◽  
Systematic Study ◽  
Free Groups ◽  
Lower Central Series ◽  
Central Series

AbstractWe make a systematic study of filtrations of a free group

ON TORSION IN FACTORS OF POLYNILPOTENT SERIES OF A GROUP WITH A SINGLE RELATION

2004 ◽  
Vol 14 (04) ◽  
pp. 513-523 ◽  
Author(s):  
C. K. GUPTA ◽  
N. S. ROMANOVSKI
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Lower Central Series ◽  
Central Series ◽  
Defining Relation ◽  
Proper Power ◽  
Torsion Free ◽  
Single Relation ◽  
Central Factors

Let G=F/rF be a group with a single defining relation, r∈Fkm\Fk,m+1, Fij the term of some polynilpotent series of the free group F. We prove: the factors of the corresponding polynilpotent series of the group G are torsion free if and only if r is not a proper power of any element of F modulo Fk,m+1. We also give a description of the lower central series of a group F/[R,R] when F/R is a nilpotent group with torsion free lower central factors.

FREE GROUPS OF OUTER COMMUTATOR VARIETIES OF GROUPS

2001 ◽  
Vol 64 (2) ◽  
pp. 423-435
Author(s):  
DANIEL P. GROVES
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Lie Ring ◽  
Lie Rings ◽  
Outer Commutator ◽  
Torsion Free ◽  
Group 1

If F is a free group, 1 < i [les ] j [les ] 2i and i [les ] k [les ] i + j + 1 then F/[γj(F), γi(F), γk(F)] is residually nilpotent and torsion-free. This result is extended to 1 < i [les ] j [les ] 2i and i [les ] k [les ] 2i + 2j. It is proved that the analogous Lie rings, L/[Lj, Li, Lk] where L is a free Lie ring, are torsion-free. Candidates are found for torsion in L/[Lj, Li, Lk] whenever k is the least of {i, j, k}, and the existence of torsion in L/[Lj, Li, Lk] is proved when i, j, k [les ] 5 and k is the least of {i, j, k}.

scholarly journals On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

COMPUTING HALL EXPONENTS IN THE FREE GROUP

1993 ◽  
Vol 03 (03) ◽  
pp. 275-294 ◽  
Author(s):  
GUY MELANÇON ◽  
CHRISTOPHE REUTENAUER
Keyword(s):  
Free Group ◽  
Lower Central Series ◽  
Central Series ◽  
Linear Combinations ◽  
Nous Montrons

Nous donnons une généralisation de la décomposition de M. Hall des éléments du groupe libre en produits décroissant de commutateurs de Hall. Nous généralisons les identités de Thérien, qui expriment les exposants de la décomposition comme des sommes à coefficients entiers positifs de fonctions sous-mots. Nous étudions l’algèbre des fonctions sous-mots et nous montrons que cette algèbre est librement engendrée par les fonctions qui donnent ces exposants; nous montrons aussi la continuité de ces fonctions pour la topologie de Hall sur le groupe libre. De plus, nous donnons de nouvelles preuves de résultats connus, entre autres les théorèmes de Magnus et Witt qui caractérisent les éléments de la série centrale descendante du grouple libre. We give the generalization of M. Hall’s expansion of each element of the free group as a decreasing product of Hall commutators. We also prove the generalization of Therien’s identities expressing the Hall exponents as nonnegative linear combinations of subword functions. We study the algebra of subword functions and show that it is freely generated by the Hall exponents functions; we also prove the continuity of these functions for the Hall topology on the free group. Besides these results, we give new proofs of known results, especially of the theorem of Magnus and Witt on the lower central series of the free group.

scholarly journals Dimension quotients of metabelian Lie rings

2017 ◽  
Vol 27 (02) ◽  
pp. 251-258
Author(s):  
Inder Bir S. Passi ◽  
Thomas Sicking
Keyword(s):  
Lower Central Series ◽  
Universal Enveloping Algebra ◽  
Central Series ◽  
Augmentation Ideal ◽  
Lie Ring ◽  
Ring Of Integers ◽  
Enveloping Algebra ◽  
Lie Rings

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].

scholarly journals On the length of the shortest non-trivial element in the derived and the lower central series

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Abdelrhman Elkasapy ◽  
Andreas Thom
Keyword(s):  
Lower Bounds ◽  
Free Group ◽  
Lower Central Series ◽  
Upper And Lower Bounds ◽  
Compact Groups ◽  
Central Series ◽  
Residual Finiteness ◽  
Trivial Element ◽  
Nilpotent Residual ◽  
Derived Series

AbstractWe provide upper and lower bounds on the length of the shortest non-trivial element in the derived series and lower central series in the free group on two generators. The techniques are used to provide new estimates on the nilpotent residual finiteness growth and on almost laws for compact groups.

A Note on Torsion Free Groups Generated by Pairs of Matrices

1975 ◽  
Vol 17 (5) ◽  
pp. 747-748 ◽  
Author(s):  
A. Charnow
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Complex Numbers ◽  
Torsion Free

Let and let Gm be the group generated by A and the transpose of A. The problem of determining complex numbers m such that Gm is a free group had been studied by several authors [1, 2, 3]. In this note we characterize those rational values of m for which Gm is torsion free.

More characteristic subgroups, Lie rings, and isomorphism tests for p-groups

2013 ◽  
Vol 16 (6) ◽  
Author(s):  
James B. Wilson
Keyword(s):  
Lower Central Series ◽  
Central Series ◽  
Arbitrary Length ◽  
Verbal Subgroup ◽  
Lie Rings

Abstract.We introduce three families of characteristic subgroups that refine the traditional verbal subgroup filters, such as the lower central series, to an arbitrary length. We prove that a positive logarithmic proportion of finite

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