CHARACTERISTIC RELATIONS FOR A FINITE-BY-NILPOTENT GROUP

2005 ◽  
Vol 15 (02) ◽  
pp. 273-277
Author(s):  
GÉRARD ENDIMIONI
Keyword(s):  
Positive Integer ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Finitely Generated

We improve previous results by showing that a finitely generated soluble group G is finite-by-nilpotent if and only if for all a, b ∈ G, there exists a positive integer n such that [a, nb] belongs to γn+2(<a, b>).<a<a,b>>′.

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 Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

scholarly journals Characterisation of nilpotent-by-finite groups

2000 ◽  
Vol 61 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Nadir Trabelsi
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Soluble Group ◽  
Finitely Generated ◽  
Positive Integers

LetGbe a finitely generated soluble group. The main result of this note is to prove thatGis nilpotent-by-finite if, and only if, for every pairX,Yof infinite subsets ofG, there exist anxinX,yinYand two positive integersm=m(x,y),n=n(x,y) satisfying [x,nym] = 1. We prove also that ifGis infinite and ifmis a positive integer, thenGis nilpotent-by-(finite of exponent dividingm) if, and only if, for every pairX,Yof infinite subsets ofG, there exist anxinX,yinYand a positive integern=n(x,y) satisfying [x,nym] = 1.

Centralizer of Engel Elements in a Group

2010 ◽  
Vol 17 (03) ◽  
pp. 487-494 ◽  
Author(s):  
Gérard Endimioni ◽  
Carmela Sica
Keyword(s):  
Nilpotent Group ◽  
General Result ◽  
Soluble Group ◽  
Finitely Generated ◽  
Radical Group ◽  
Engel Elements ◽  
Černikov Group ◽  
Polycyclic Groups ◽  
Finiteness Properties

In this paper we show that some finiteness properties on a centralizer of a particular subgroup can be inherited by the whole group. Among other things, we prove the following characterization of polycyclic groups: a soluble group G is polycyclic if and only if it contains a finitely generated subgroup H, formed by bounded left Engel elements, whose centralizer CG(H) is polycyclic. In the context of Černikov groups we obtain a more general result: a radical group is a Černikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Černikov group. The aforementioned results generalize a theorem by Onishchuk and Zaĭtsev about the centralizer of a finitely generated subgroup in a nilpotent group.

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.

scholarly journals A question of Paul Erdös and nilpotent-by-finite groups

2001 ◽  
Vol 64 (2) ◽  
pp. 245-254
Author(s):  
Bijan Taeri
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Soluble Group ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Finite Quotient

Let n be a positive integer or infinity (denoted ∞), k a positive integer. We denote by Ωk(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist distinct elements x, y ∈ X and integers t0, t1…, tk such that , where xi, ∈ {x, y}, i = 0, 1,…,k, x0 ≠ x1. If the integers t0, t1,…,tk are the same for any subset X of G, we say that G is in the class Ω̅k(n). The class k (n) is defined exactly as Ωk(n) with the additional conditions . Let t2, t3,…,tk be fixed integers. We denote by the class of all groups G such that for any infinite subsets X and Y there exist x ∈ X, y ∈ Y such that , where xi ∈ {x, y}, x0 ≠ x1, i = 2, 3, …, k. Here we prove that (1) If G ∈ k(2) is a finitely generated soluble group, then G is nilpotent.(2) If G ∈ Ωk(∞) is a finitely generated soluble group, then G is nilpotentby-finite.(3) If G ∈ Ω̅k(n), n a positive integer, is a finitely generated residually finite group, then G is nilpotent-by-finite.(4) If G is an infinite -group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient, then G is nilpotent-by-finite.

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.

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

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 Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

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