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.

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 BOGOMOLOV MULTIPLIERS OF P-GROUPS OF MAXIMAL CLASS

2019 ◽  
Vol 71 (1) ◽  
pp. 123-138
Author(s):  
Gustavo A FernÁndez-Alcober ◽  
Urban Jezernik
Keyword(s):  
Lower Central Series ◽  
Maximal Class ◽  
Central Series ◽  
Natural Family ◽  
Positive Degree ◽  
Bogomolov Multiplier

Abstract Let $G$ be a $p$-group of maximal class and order $p^n$. We determine whether or not the Bogomolov multiplier ${\operatorname{B}}_0(G)$ is trivial in terms of the lower central series of $G$ and $P_1 = C_G(\gamma _2(G) / \gamma _4(G))$. If in addition $G$ has positive degree of commutativity and $P_1$ is metabelian, we show how understanding ${\operatorname{B}}_0(G)$ reduces to the simpler commutator structure of $P_1$. This result covers all $p$-groups of maximal class of large-enough order, and, furthermore, it allows us to give the first natural family of $p$-groups containing an abundance of groups with non-trivial Bogomolov multipliers. We also provide more general results on Bogomolov multipliers of $p$-groups of arbitrary coclass $r$.

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.

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.

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.

Some remarks on subgroups of maximal class in a finite p-group

2015 ◽  
Vol 14 (04) ◽  
pp. 1550043
Author(s):  
Yakov Berkovich
Keyword(s):  
Normal Subgroup ◽  
Lower Central Series ◽  
Maximal Class ◽  
Central Series ◽  
Minimal Order ◽  
Epimorphic Image ◽  
Class C ◽  
Class Iv

The following characterizations of p-groups of maximal class are proved: (a) If a p-group of order > pp+2 contains a subgroup of maximal class and index p, then G possesses at most one normal subgroup of order pp and exponent p. (b) If the center of any nonabelian epimorphic image of a nonabelian two-generator p-group G is cyclic, then either G ≅ M pn or G is of maximal class. (c) An 𝒜n-group G, n > 1, is of maximal class ⇔ all its 𝒜2-subgroups of minimal order are of maximal class. (iv) If all factors of the lower central series of a nonabelian two-generator p-group are cyclic, then it is either of maximal class or ≅ M pn. (v) If a nonabelian p-group G is such that any s pairwise non-commuting elements generate a group of maximal class, where s is the fixed member of the set {3, …, p + 1} and p > 2 if s ≠ p + 1, then G is also of maximal class. We also study the noncyclic p-groups containing only one normal subgroup of a given order.

scholarly journals On Central Series

1962 ◽  
Vol 13 (2) ◽  
pp. 175-178 ◽  
Author(s):  
I. D. Macdonald
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Lower Central Series ◽  
The Other ◽  
Central Series ◽  
Image Position ◽  
A Finite Group ◽  
Positive Integers

Letandbe, respectively, the upper and lower central series of a group G. Our purpose in this note is to extend known results and find some information as to which of the factors Zk/Zk−1 and Γk/Γk+1 may be infinite. Though our conclusions about the lower central series will be quite general we assume in the other case that the group is f.n., i.e. an extension of a finite group by a nilpotent group. The essential facts about f.n. groups are to be found in P. Hall's paper (4). We also refer to (4) for general notation; we reserve the letter k for positive integers.

scholarly journals Generation of the lower central series

1982 ◽  
Vol 23 (1) ◽  
pp. 15-20 ◽  
Author(s):  
Robert M. Guralnick
Keyword(s):  
Lower Central Series ◽  
Central Series ◽  
Image Position

Let G be a group. The rth term LrG of the lower central series of G is the subgroup generated by the r-fold commutatorswhere , and for r > 1,

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 (*)

On prime-power groups with two generators

1958 ◽  
Vol 54 (3) ◽  
pp. 327-337 ◽  
Author(s):  
N. Blackburn
Keyword(s):  
Prime Power ◽  
Lower Central Series ◽  
Central Series ◽  
Frattini Subgroup ◽  
Number Of Generators ◽  
Image Position

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be writtenFor any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.

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