scholarly journals Quasivarieties of nilpotent groups of axiomatic rank $4$

2020 ◽  
Vol 17 ◽  
pp. 2131-2141
Author(s):  
A. I. Budkin
Keyword(s):  
Nilpotent Groups ◽  
Axiomatic Rank

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}}.

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.

scholarly journals Equations in nilpotent groups

2015 ◽  
Vol 143 (11) ◽  
pp. 4723-4731 ◽  
Author(s):  
Moon Duchin ◽  
Hao Liang ◽  
Michael Shapiro
Keyword(s):  
Nilpotent Groups
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