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.

On Certain Properties of Subnormal Subgroups

1978 ◽  
Vol 30 (03) ◽  
pp. 573-582 ◽  
Author(s):  
Jennifer Whitehead
Keyword(s):  
Lower Central Series ◽  
Nilpotent Groups ◽  
Composition Series ◽  
Central Series ◽  
Image Position ◽  
Subnormal Subgroups

Main results. Let G be a group generated by two subnormal subgroups H and K. Denoting the class of nilpotent groups by 𝔑, and the limit of the lower central series by G𝔑, Wielandt showed in [14], for groups with a composition series that (*)

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.

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.

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.

scholarly journals Longer nilpotent series for classical unipotent subgroups

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Maglione
Keyword(s):  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series

AbstractIn studying nilpotent groups, the lower central series and other variations can be used to construct an associated ℤ

Classification of p-groups of automorphisms of Riemann surfaces and their lower central series

1987 ◽  
Vol 29 (2) ◽  
pp. 237-244 ◽  
Author(s):  
Reza Zomorrodian
Keyword(s):  
Nilpotent Group ◽  
Riemann Surfaces ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Fuchsian Groups ◽  
Central Series ◽  
Local Form ◽  
Precise Information ◽  
Groups Of Automorphisms ◽  
Hurwitz Theorem

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local formThe restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.

scholarly journals Some automorphisms of finite nilpotent groups

1960 ◽  
Vol 4 (4) ◽  
pp. 204-207 ◽  
Author(s):  
J. C. Howarth
Keyword(s):  
Finite Groups ◽  
Lower Central Series ◽  
Nilpotent Groups ◽  
Central Series ◽  
Endomorphism Semigroup ◽  
Group Of Automorphisms ◽  
Inner Automorphism ◽  
Group A ◽  
Special Case

This note extends the concept of the inner automorphism, but here applies only to those finite groups G for which some member of the lower central series is Abelian. In general (e.g. when G is metabelian) the construction yields an endomorphism semigroup, but in the special case where Gis nilpotent (and may therefore, for our present purposes, be considered as a p-group) a group of automorphisms results.

On nilpotency of the group of outer class-preserving automorphisms of a group

2015 ◽  
Vol 15 (02) ◽  
pp. 1650026 ◽  
Author(s):  
Pradeep K. Rai
Keyword(s):  
Lower Central Series ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Trivial Group ◽  
Improve Bound

Let G be a group and Zj(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Out c(G/Zj(G)), the group of outer class-preserving automorphisms of G/Zj(G), is nilpotent of class k, then Out c(G) is nilpotent of class at most j + k. Moreover, if Out c(G/Zj(G)) is a trivial group, then Out c(G) is nilpotent of class at most j. As an application we prove that if γi(G)/γi(G) ∩ Zj(G) is cyclic then Out c(G) is nilpotent of class at most i + j, where γi(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Out c(G) for some classes of nilpotent groups G.

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