scholarly journals On quasinormal subgroups of certain finitely generated groups

1983 ◽  
Vol 26 (1) ◽  
pp. 25-28 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Nilpotent Group ◽  
Normal Closure ◽  
Infinite Index ◽  
Finitely Generated ◽  
The Core ◽  
Finitely Generated Groups ◽  
Quasinormal Subgroup ◽  
Finite Exponent

A subgroup Q of a group G is called quasinormal in G if Q permutes with every subgroup of G. Of course a quasinormal subgroup Q of a group G may be very far from normal. In fact, examples of Iwasawa show (for a convenient reference see [8]) that we may have Q core-free and the normal closure QG of Q in G equal to G so that Q is not even subnormal in G. We note also that the core of Q in G, QG, is of infinite index in QG in this example. If G is finitely generated then any quasinormal subgroup Q is subnormal in G [8] and although Q is not necessarily normal in G we have that |QG:Q| is finite and |QG:Q| is a nilpotent group of finite exponent [5].

scholarly journals Group Laws Implying Virtual Nilpotence

2003 ◽  
Vol 74 (3) ◽  
pp. 295-312 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Additional Condition ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Locally Graded Groups ◽  
Finite Exponent ◽  
Group Law

AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

scholarly journals ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL

2012 ◽  
Vol 22 (05) ◽  
pp. 1250048 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
JOSÉ BURILLO ◽  
MURRAY ELDER ◽  
ZORAN ŠUNIĆ
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
The Other ◽  
Normal Closure ◽  
Finitely Generated ◽  
Generating Set ◽  
Finitely Generated Group ◽  
Cyclic Groups ◽  
Growth Functions ◽  
Generating Sets

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

Characterization of finitely generated groups by types

2018 ◽  
Vol 28 (08) ◽  
pp. 1613-1632 ◽  
Author(s):  
A. G. Myasnikov ◽  
N. S. Romanovskii
Keyword(s):  
Free Surface ◽  
Nilpotent Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Burnside Groups ◽  
Text Types ◽  
Rigid Group

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups [Formula: see text] are fully characterized in the class of all groups by the set [Formula: see text] of types realized in [Formula: see text]. Furthermore, it turns out that these groups [Formula: see text] are fully characterized already by some particular rather restricted fragments of the types from [Formula: see text]. In particular, every finitely generated nilpotent group is completely defined by its [Formula: see text]-types, while a finitely generated rigid group is completely defined by its [Formula: see text]-types, and a finitely generated metabelian or polycyclic group is completely defined by its [Formula: see text]-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

On Metanilpotent Varieties of Groups

1970 ◽  
Vol 22 (4) ◽  
pp. 875-877 ◽  
Author(s):  
Narain Gupta
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finite Exponent ◽  
Torsion Free

Let denote the variety of all groups which are extensions of a nilpotent-of-class-c group by a nilpotent-of-class-d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem.THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group.This generalizes the main theorem of Ŝmel′kin [4], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [2] for a related discussion.

scholarly journals Condensed groups in product varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Denis Osin
Keyword(s):  
Isomorphism Class ◽  
Product Variety ◽  
Equivalence Relations ◽  
Finitely Generated ◽  
Locally Finite ◽  
Elementary Equivalence ◽  
Finitely Generated Groups ◽  
Locally Finite Variety ◽  
Isolated Points ◽  
Finite Exponent

Abstract A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U ⁢ V \mathcal{UV} , where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in U ⁢ V \mathcal{UV} .

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.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

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.

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