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

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.

Augmentation quotients of some nonabelian finite groups

1979 ◽  
Vol 85 (2) ◽  
pp. 261-270 ◽  
Author(s):  
Gerald Losey ◽  
Nora Losey
Keyword(s):  
Group Structure ◽  
Lower Central Series ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Nilpotent Residual ◽  
Periodic Behaviour ◽  
Augmentation Quotient ◽  
Image Position ◽  
A Finite Group

1. LetGbe a group,ZGits integral group ring and Δ = ΔGthe augmentation idealZGBy anaugmentation quotientofGwe mean any one of theZG-moduleswheren, r≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotientsQn(G) =Qn,1(G) and(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determineQn(G) andPn(G) for finiteGit is sufficient to assume thatGis ap-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelianp-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: ifGis a finite group then there exist natural numbersn0and π such thatQn(G) ≅Qn+π(G) for alln≥n0; ifGωis the nilpotent residual ofGandG/Gωhas classcthen π divides l.c.m. {1, 2, …,c}. There do not appear to be any examples in the literature of this periodic behaviour forc> 1. One of goals here is to present such examples. These examples will be from the class of finitep-groups in which the lower central series is anNp-series.

Finite-by-nilpotent groups

1956 ◽  
Vol 52 (4) ◽  
pp. 611-616 ◽  
Author(s):  
P. Hall
Keyword(s):  
Commutator Subgroup ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series ◽  
Image Position

1. Introduction. 1·1. Notation. Letandbe, respectively, the upper and lower central series of a group G. By definition, Zi+1/Zi is the centre of G/Zi and Γj+1 = [Γj, G] is the commutator subgroup of Γj with G. When necessary for clearness, we write ZiG) for Zi and Γj(G) for Γj.

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 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.

Augmentation quotients of group rings and symmetric powers

1979 ◽  
Vol 85 (2) ◽  
pp. 247-252 ◽  
Author(s):  
Robert Sandling ◽  
Ken-Ichi Tahara
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Additive Group ◽  
Lower Central Series ◽  
Symmetric Power ◽  
Central Series ◽  
Group Rings ◽  
Augmentation Ideal ◽  
Symmetric Powers ◽  
Image Position

Let G be a group with the lower central seriesLetwhere Σ runs over all non-negative integers a1, a2,…, an such that and is the aith symmetric power of the abelian group Gi/Gi+1 whereLet I (G) be the augmentation ideal of G in , the group ring of G over . Define the additive group Qn (G) = In (G) / In+1 (G) for any n ≥ 1. Then it is well known that Q1(G) ≅ W1(G) for any group G. Losey (4,5) proved that Q2(G) ≅ W2(G) for any finitely generated group G. Furthermore recently Tahara(12) proved that Q3(G) is a certain precisely defined quotient of W3(G) for any finite group G.

scholarly journals 2-groups of almost maximal class

1975 ◽  
Vol 19 (3) ◽  
pp. 343-357 ◽  
Author(s):  
Rodney James
Keyword(s):  
Lower Central Series ◽  
Maximal Class ◽  
Central Series ◽  
Image Position

Let G be a group of order 2n and x, y ∈ G. We define the Commutator [x, y] as x−1y−1xy. Similarly, if X, Y are subsets of G, then [X, Y] denotes the sub-group genrated by all commutators of the form [x, y] where x ∈ X, y ∈ Y. Using this, we may define the lower central series of G inductively by The following results are well known.

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