On connection between nilpotent groups and Lie rings

2000 ◽  
Vol 41 (5) ◽  
pp. 994-1004 ◽  
Author(s):  
A. Jaikin Zapirain ◽  
E. I. Khukhro
Keyword(s):  
Nilpotent Groups ◽  
Lie Rings

scholarly journals IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS

2020 ◽  
Vol 71 (3) ◽  
pp. 959-980
Author(s):  
Christopher Voll
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Infinite Family ◽  
Nilpotent Groups ◽  
Unipotent Group ◽  
Group Schemes ◽  
Lie Rings ◽  
Local Functional ◽  
Explicit Formulae ◽  
Local Functional Equations

Abstract We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.

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.

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.

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.

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}}.

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.

Powerfully Nilpotent Groups of Class 2

2021 ◽  
pp. 1-16
Author(s):  
Jason Semeraro
Keyword(s):  
Nilpotent Groups

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

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