scholarly journals Powerfully nilpotent groups of maximal powerful class

2019 ◽  
Vol 191 (4) ◽  
pp. 779-799
Author(s):  
G. Traustason ◽  
J. Williams
Keyword(s):  
Positive Integer ◽  
Nilpotent Group ◽  
Special Kind ◽  
Nilpotent Groups ◽  
Maximal Class ◽  
Central Series ◽  
Lie Ring ◽  
Lie Rings ◽  
Lazard Correspondence

Abstract In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime $$p>r$$p>r, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.

Powerfully nilpotent groups of rank 2 or small order

2020 ◽  
Vol 23 (4) ◽  
pp. 641-658
Author(s):  
Gunnar Traustason ◽  
James Williams
Keyword(s):  
Detailed Analysis ◽  
Special Kind ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Rank 2 ◽  
Full Classification

AbstractIn this paper, we continue the study of powerfully nilpotent groups. These are powerful p-groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank 2. The classification will then be used to arrive at a precise formula for the number of powerfully nilpotent groups of rank 2 and order {p^{n}}. We will also give a detailed analysis of the ancestry tree for these groups. The second part of the paper is then devoted to a full classification of powerfully nilpotent groups of order up to {p^{6}}.

scholarly journals Dimension quotients of metabelian Lie rings

2017 ◽  
Vol 27 (02) ◽  
pp. 251-258
Author(s):  
Inder Bir S. Passi ◽  
Thomas Sicking
Keyword(s):  
Lower Central Series ◽  
Universal Enveloping Algebra ◽  
Central Series ◽  
Augmentation Ideal ◽  
Lie Ring ◽  
Ring Of Integers ◽  
Enveloping Algebra ◽  
Lie Rings

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].

Some Engel conditions on infinite subsets of certain groups

2000 ◽  
Vol 62 (1) ◽  
pp. 141-148 ◽  
Author(s):  
Alireza Abdollahi
Keyword(s):  
Positive Integer ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Metabelian Group ◽  
Infinite Subset ◽  
Central Series ◽  
Finitely Generated ◽  
Locally Nilpotent ◽  
Finite Quotient

Let k be a positive integer. We denote by ɛk(∞) the class of all groups in which every infinite subset contains two distinct elements x, y such that [x,k y] = 1. We say that a group G is an -group provided that whenever X, Y are infinite subsets of G, there exists x ∈ X, y ∈ Y such that [x,k y] = 1. Here we prove that:(1) If G is a finitely generated soluble group, then G ∈ ɛ3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3.(2) If G is a finitely generated metabelian group, then G ∈ ɛk(∞) if and only if G/Zk (G) is finite, where Zk (G) is the (k + 1)-th term of the upper central series of G.(3) If G is a finitely generated soluble ɛk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt (G) is finite.(4) If G is an infinite -group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.

scholarly journals FREE CENTRE-BY-NILPOTENT-BY-ABELIAN LIE RINGS OF RANK 2

2015 ◽  
Vol 102 (1) ◽  
pp. 63-73 ◽  
Author(s):  
MARIA ALEXANDROU ◽  
RALPH STÖHR
Keyword(s):  
Free Abelian Group ◽  
Torsion Group ◽  
Lower Central Series ◽  
Torsion Subgroup ◽  
Central Series ◽  
Lie Ring ◽  
Direct Sum ◽  
Lie Rings ◽  
Rank 2 ◽  
Derived Series

We study the free Lie ring of rank $2$ in the variety of all centre-by-nilpotent-by-abelian Lie rings of derived length $3$. This is the quotient $L/([\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime })$ with $c\geqslant 2$ where $L$ is the free Lie ring of rank $2$, $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })$ is the $c$th term of the lower central series of the derived ideal $L^{\prime }$ of $L$, and $L^{\prime \prime \prime }$ is the third term of the derived series of $L$. We show that the quotient $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })+L^{\prime \prime \prime }/[\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime }$ is a direct sum of a free abelian group and a torsion group of exponent $c$. We exhibit an explicit generating set for the torsion subgroup.

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 A note on p-groups with power automorphisms

1992 ◽  
Vol 34 (3) ◽  
pp. 327-332 ◽  
Author(s):  
R. A. Bryce ◽  
John Cossey ◽  
E. A. Ormerod
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Central Series ◽  
Related Concept ◽  
Finite Nilpotent Group ◽  
Basic Properties

Let G be a group. The norm, or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G. In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω(G). The Wielandt series of subgroups ω1(G) is defined by: ω1(G) = ω(G) and for i ≥ 1, ωi+1(G)/ ω(G) = ω(G/ωi, (G)). The subgroups of the upper central series we denote by ζi(G).

scholarly journals SOME VARIETIES OF LIE RINGS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250051
Author(s):  
Yin Chen ◽  
Runxuan Zhang
Keyword(s):  
Nilpotent Groups ◽  
Lie Ring ◽  
Lie Rings

In this paper, several theorems of Macdonald [On certain varieties of groups, Math. Z.76 (1961) 270–282; On certain varieties of groups II, Math. Z.78 (1962) 175–188] on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some equations (see Eqs. (1.1)–(1.3)). We prove that each Lie ring in variety (1.1) is nilpotent of exponent at most n + 2; if L is a Lie ring in variety (1.2), then L2 is nilpotent of exponent at most n + 1; and each Lie ring in variety (1.3) is solvable of length at most n + 1. We also discuss some varieties of solvable Lie rings and the varieties of Lie rings defined by the properties of subrings.

The q-tensor square of finitely generated nilpotent groups, q odd

2017 ◽  
Vol 16 (11) ◽  
pp. 1750211 ◽  
Author(s):  
Noraí R. Rocco ◽  
Eunice C. P. Rodrigues
Keyword(s):  
Positive Integer ◽  
Nilpotent Group ◽  
Upper Bound ◽  
Finite Rank ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Number Of Generators ◽  
Tensor Square

In the present paper, the authors extend to the [Formula: see text]-tensor square [Formula: see text] of a group [Formula: see text], [Formula: see text] an odd positive integer, some structural results due to Blyth, Fumagalli and Morigi concerning the non-abelian tensor square [Formula: see text] ([Formula: see text]). The results are applied to the computation of [Formula: see text] for finitely generated nilpotent groups [Formula: see text], specially for free nilpotent groups of finite rank. We also generalize to all [Formula: see text] results of Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square [Formula: see text] when [Formula: see text] is a [Formula: see text]-generator nilpotent group of class 2. We end by computing the [Formula: see text]-tensor squares of the free [Formula: see text]-generator nilpotent group of class 2, [Formula: see text]. This shows that the above mentioned upper bound is also achieved for these groups when [Formula: see text] odd.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

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