scholarly journals Generation of the lower central series II

1984 ◽  
Vol 25 (2) ◽  
pp. 193-201 ◽  
Author(s):  
Robert M. Guralnick
Keyword(s):  
Positive Integer ◽  
Lower Central Series ◽  
Central Series ◽  
Sylow Subgroups ◽  
Image Position

In this article, we obtain results on commutators in Sylow subgroups of the lower central series, extending the work of Dark and Newell [2], Rodney [12, 13] and Aschbacher and the author [1, 6, 7].Some notation is required for the statement of the main results. Let r be a positive integer and defineandwhere x1, …, xr, are elements in a group G. Let ΓrG = {[x1, …, xr]∣ x1 ∈ G} be the set of r-fold commutators in G. Then Lr,G = 〈ΓrG〉 is the rth term in the lower central series of G. Set L∞G = ∩ Lr,G.

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.

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.

BOUNDING THE ORDER OF THE NILPOTENT RESIDUAL OF A FINITE GROUP

2016 ◽  
Vol 94 (2) ◽  
pp. 273-277
Author(s):  
AGENOR FREITAS DE ANDRADE ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Lower Central Series ◽  
Central Series ◽  
Nilpotent Residual ◽  
A Finite Group

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.

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.

New Facts about the Vanishing Off Subgroup

2019 ◽  
Vol 63 (2) ◽  
pp. 262-268
Author(s):  
Nabil Mlaiki
Keyword(s):  
Positive Integer ◽  
Lower Central Series ◽  
Central Series

AbstractIn this manuscript, we generalize Lewis’s result about a central series associated with the vanishing off subgroup. We write $V_{1}=V(G)$ for the vanishing off subgroup of $G$, and $V_{i}=[V_{i-1},G]$ for the terms in this central series. Lewis proved that there exists a positive integer $n$ such that if $V_{3}<G_{3}$, then $|G\,:\,V_{1}|=|G^{\prime }\,:\,V_{2}|^{2}=p^{2n}$. Let $D_{3}/V_{3}=C_{G/V_{3}}(G^{\prime }/V_{3})$. He also showed that if $V_{3}<G_{3}$, then either $|G\,:\,D_{3}|=p^{n}$ or $D_{3}=V_{1}$. We show that if $V_{i}<G_{i}$ for $i\geqslant 4$, where $G_{i}$ is the $i$-th term in the lower central series of $G$, then $|G_{i-1}\,:\,V_{i-1}|=|G\,:\,D_{3}|$.

Groups with Finiteness Conditions on the Lower Central Series of Subgroups

2013 ◽  
Vol 20 (04) ◽  
pp. 663-670 ◽  
Author(s):  
Silvana Rinauro
Keyword(s):  
Positive Integer ◽  
Lower Central Series ◽  
Central Series ◽  
Finiteness Conditions ◽  
Locally Graded Groups

Let k be a positive integer. Locally graded groups G for which one of the sets {γk(H)| H ≤ G} and {γk(H)| H ≤G, H infinite } is finite are classified.

Groups with a Certain Condition on Conjugates

1952 ◽  
Vol 4 ◽  
pp. 369-372 ◽  
Author(s):  
Franklin Haimo
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Lower Central Series ◽  
Central Series ◽  
Uniform Bound

In this paper, we shall show that if is a nilpotent [5] group and if M, a positive integer, is a uniform bound on the number of conjugates that any element of may have, then there exist “large” integers n for which x → xn is a central endomorphism of . If is not necessarily nilpotent, if the above condition on the conjugates is retained, and if we can find a member of the lower central series [1], every element of which lies in some member of the ascending central series, then we shall show that every non-unity element of the “high” derivatives has finite order.

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