Universal theory of the class of finite nilpotent groups is unsolvable

1983 ◽  
Vol 33 (4) ◽  
pp. 254-263
Author(s):  
O. G. Kharlampovich
Keyword(s):  
Nilpotent Groups ◽  
Universal Theory

∀-free metabelian groups

1997 ◽  
Vol 62 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Olivier Chapuis
Keyword(s):  
Free Group ◽  
Metabelian Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Explicit Description ◽  
Universal Theory ◽  
Metabelian Groups ◽  
Free Metabelian Group ◽  
Finitely Generated Groups

In 1965, during the first All-Union Symposium on Group Theory, Kargapolov presented the following two problems: (a) describe the universal theory of free nilpotent groups of class m; (b) describe the universal theory of free groups (see [18, 1.28 and 1.27]). The first of these problems is still open and it is known [25] that a positive solution of this problem for an m ≤ 2 should imply the decidability of the universal theory of the field of the rationals (this last problem is equivalent to Hilbert's tenth problem for the field of the rationals which is a difficult open problem; see [17] and [20] for discussions on this problem). Regarding the second problem, Makanin proved in 1985 that a free group has a decidable universal theory (see [15] for stronger results), however, the problem of deriving an explicit description of the universal theory of free groups is open. To try to solve this problem Remeslennikov gave different characterization of finitely generated groups with the same universal theory as a noncyclic free group (see [21] and [22] and also [11]). Recently, the author proved in [8] that a free metabelian group has a decidable universal theory, but the proof of [8] does not give an explicit description of the universal theory of free metabelian groups.

Universal theory of nilpotent groups

1979 ◽  
Vol 25 (4) ◽  
pp. 253-258 ◽  
Author(s):  
V. A. Roman'kov
Keyword(s):  
Nilpotent Groups ◽  
Universal Theory

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.

Synergetic Theory of Document in Evolution of Document Science

2018 ◽  
pp. 1060-1068
Author(s):  
Galina A. Dvoenosova ◽  
Keyword(s):  
Social Role ◽  
Scientific Study ◽  
Self Organization ◽  
Universal Theory ◽  
Social Self ◽  
Social Goal ◽  
The Social ◽  
Synergetic Theory ◽  
First Time

The article assesses synergetic theory of document as a new development in document science. In information society the social role of document grows, as information involves all members of society in the process of documentation. The transformation of document under the influence of modern information technologies increases its interest to representatives of different sciences. Interdisciplinary nature of document as an object of research leads to an ambiguous interpretation of its nature and social role. The article expresses and contends the author's views on this issue. In her opinion, social role of document is incidental to its being a main social tool regulating the life of civilized society. Thus, the study aims to create a scientific theory of document, explaining its nature and social role as a tool of social (goal-oriented) action and social self-organization. Substantiation of this idea is based on application of synergetics (i.e., universal theory of self-organization) to scientific study of document. In the synergetic paradigm, social and historical development is seen as the change of phases of chaos and order, and document is considered a main tool that regulates social relations. Unlike other theories of document, synergetic theory studies document not as a carrier and means of information transfer, but as a unique social phenomenon and universal social tool. For the first time, the study of document steps out of traditional frameworks of office, archive, and library. The document is placed on the scales with society as a global social system with its functional subsystems of politics, economy, culture, and personality. For the first time, the methods of social sciences and modern sociological theories are applied to scientific study of document. This methodology provided a basis for theoretical vindication of nature and social role of document as a tool of social (goal-oriented) action and social self-organization. The study frames a synergetic theory of document with methodological foundations and basic concepts, synergetic model of document, laws of development and effectiveness of document in the social continuum. At the present stage of development of science, it can be considered the highest form of theoretical knowledge of document and its scientific explanatory theory.

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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