Natural Boundaries III: Nilpotent Groups

2008 ◽  
pp. 169-177
Keyword(s):  
Nilpotent Groups ◽  
Natural Boundaries

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}}.

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.

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

scholarly journals The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

1992 ◽  
Vol 53 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Raptis ◽  
D. Varsos
Keyword(s):  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Hnn Extensions ◽  
Base Group ◽  
Generalized Free Products

AbstractWe study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

scholarly journals Short- and long-term movement patterns of six temperate reef fishes (Families Labridae and Monacanthidae)

1995 ◽  
Vol 46 (5) ◽  
pp. 853 ◽  
Author(s):  
NS Barrett
Keyword(s):  
Natural Habitat ◽  
Movement Patterns ◽  
Reef Fishes ◽  
Long Term Results ◽  
Home Ranges ◽  
Temperate Reef ◽  
Effective Deterrent ◽  
Short And Long Term ◽  
Natural Boundaries

Movement patterns were studied on a 1-ha isolated reef surrounding Arch Rock in southern Tasmania. Short-term movements were identified from diver observations, and interpretation of long-term movements involved multiple recaptures of tagged individuals. Visual observations indicated that the sex-changing labrids Notolabrus tetricus, Pictilabrus laticlavius and Pseudolabrus psittaculus were all site-attached, with females having overlapping home ranges and males being territorial. In the non-sex-changing labrid Notolabrus fucicola and in the monacanthids Penicipelta vittiger and Meuschenia australis, there was no evidence of territorial behaviour and 1-h movements were in excess of the scale of the study. The long-term results indicated that all species were permanent reef residents, with most individuals of all species except M. australis always being recaptured within a home range of 100 m × 25 m or less. Only 15% of individuals of M. australis were always recaptured within this range category. The natural habitat boundary of open sand between the Arch Rock reef and adjacent reefs appeared to be an effective deterrent to emigration. The use of natural boundaries should be an important consideration in the design of marine reserves where the aim is to minimize the loss of protected species to adjacent fished areas.

Irreducible complex representations of finitely generated nilpotent groups

1978 ◽  
Vol 29 (4) ◽  
pp. 331-337
Author(s):  
S. D. Berman ◽  
V. V. Sharaya
Keyword(s):  
Nilpotent Groups ◽  
Finitely Generated
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