Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series

1958 ◽  
Vol 68 (1) ◽  
pp. 81 ◽  
Author(s):  
K. T. Chen ◽  
R. H. Fox ◽  
R. C. Lyndon
Keyword(s):  
Differential Calculus ◽  
Lower Central Series ◽  
Central Series ◽  
Quotient Groups

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 Quotient groups of IA-automorphisms of a free group of rank 3

2018 ◽  
Vol 28 (01) ◽  
pp. 115-131 ◽  
Author(s):  
V. Metaftsis ◽  
A. I. Papistas ◽  
I. Sevaslidou
Keyword(s):  
Lie Algebra ◽  
Free Group ◽  
Quotient Group ◽  
Lower Central Series ◽  
Central Series ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Graded Lie Algebra ◽  
Necessary And Sufficient ◽  
Quotient Groups

We prove that, for any positive integer [Formula: see text], the quotient group [Formula: see text] of the lower central series of the McCool group [Formula: see text] is isomorphic to two copies of the quotient group [Formula: see text] of the lower central series of a free group [Formula: see text] of rank [Formula: see text] as [Formula: see text]-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra [Formula: see text] of [Formula: see text] is naturally embedded into the Johnson Lie algebra [Formula: see text] of the IA-automorphisms of [Formula: see text].

scholarly journals On the lower central series quotients of a graded associative algebra

2011 ◽  
Vol 328 (1) ◽  
pp. 287-300 ◽  
Author(s):  
Martina Balagović ◽  
Anirudha Balasubramanian
Keyword(s):  
Associative Algebra ◽  
Lower Central Series ◽  
Central Series

scholarly journals Groups of the virtual trefoil and Kishino knots

2018 ◽  
Vol 27 (13) ◽  
pp. 1842009
Author(s):  
Valeriy G. Bardakov ◽  
Yuliya A. Mikhalchishina ◽  
Mikhail V. Neshchadim
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Lower Central Series ◽  
Central Series ◽  
Virtual Link ◽  
Virtual Knots ◽  
Formal Power

In the paper [13], for an arbitrary virtual link [Formula: see text], three groups [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] were defined. In the present paper, these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also, we prove that [Formula: see text] distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence, we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.

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.

scholarly journals Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

scholarly journals SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

2007 ◽  
Vol 16 (10) ◽  
pp. 1295-1329
Author(s):  
E. KALFAGIANNI ◽  
XIAO-SONG LIN
Keyword(s):  
Fundamental Group ◽  
Lower Central Series ◽  
Central Series ◽  
Seifert Surface ◽  
Vassiliev Invariants ◽  
Seifert Surfaces

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

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