scholarly journals Mal’tsev bases for partially commutative nilpotent groups

2021 ◽  
pp. 1-9
Author(s):  
E. I. Timoshenko
Keyword(s):  
Nilpotent Group ◽  
Canonical Form ◽  
Ordered Set ◽  
Nilpotent Groups ◽  
Partially Commutative Nilpotent Group

We construct an ordered set of commutators in a partially commutative nilpotent group [Formula: see text]. This set allows us to define a canonical form for each element of [Formula: see text]. Namely, we construct a Mal’tsev basis for the group [Formula: see text]

Polynomials with Certain Prescribed Conditions on the Galois Group

1969 ◽  
Vol 21 ◽  
pp. 262-273 ◽  
Author(s):  
Elizabeth Rowlinson ◽  
Hans Schwerdtfeger
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Canonical Form ◽  
Nilpotent Groups ◽  
Faithful Representation ◽  
Splitting Field ◽  
Normal Extension ◽  
Algebraic Equations ◽  
Abstract Group ◽  
Group A

In this paper, some contributions are made to the theory of algebraic equations over the rational field with conditions imposed on the Galois group. In § 1, for a given abstract group G all faithful permutation representations Ḡ are obtained, and it is shown that if one of them is the group of some equation with splitting field K, then any of them is the group of some equation, also with splitting field K. The method of proof enables us to construct an equation having as group a given faithful permutation representation Ḡ of a prescribed group G if we are given an equation having as group some faithful representation of G. In § 2, equations having nilpotent group are considered, non-normal extension fields are discussed, and a canonical form is obtained for the roots of non-normal irreducible equations; this form is used to characterize fields and equations with nilpotent groups.

The Structure of 𝐺-Free Nilpotent Groups of Step 3

2004 ◽  
Vol 11 (1) ◽  
pp. 27-33
Author(s):  
M. Amaglobeli
Keyword(s):  
Nilpotent Group ◽  
Canonical Form ◽  
Nilpotent Groups ◽  
Free Nilpotent Group

Abstract The canonical form of elements of a 𝐺-free nilpotent group of step 3 is defined assuming that the group 𝐺 contains no elements of order 2.

scholarly journals CENTRALISER DIMENSION OF FREE PARTIALLY COMMUTATIVE NILPOTENT GROUPS OF CLASS 2

2008 ◽  
Vol 50 (2) ◽  
pp. 251-269
Author(s):  
VIKKI A. BLATHERWICK
Keyword(s):  
Algebraic Geometry ◽  
Nilpotent Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Commutative Group ◽  
Partially Commutative Group ◽  
Partially Commutative Nilpotent Group

AbstractIn an effort to extend the theory of algebraic geometry over groups beyond free groups, Duncan, Kazatchkov and Remeslennikov have studied the notion of centraliser dimension for free partially commutative groups. In this paper we consider the centraliser dimension of free partially commutative nilpotent groups of class 2, showing that a free partially commutative nilpotent group of class 2 with non-commutation graph Γ has the same centraliser dimension as the free partially commutative group represented by the non-commutation graph Γ.

The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
Bettina Eick ◽  
Ann-Kristin Engel
Keyword(s):  
Nilpotent Group ◽  
Canonical Form ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Explicit Description ◽  
Isomorphism Type ◽  
Polynomial Equations ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Torsion Free

AbstractWe consider the isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.

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.

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.

Stability of nilpotent groups of class 2 and prime exponent

1981 ◽  
Vol 46 (4) ◽  
pp. 781-788 ◽  
Author(s):  
Alan H. Mekler
Keyword(s):  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Stability Class ◽  
Similarity Type ◽  
Nilpotent Class ◽  
Prime Exponent

AbstractLet p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M.Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes.Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

Genus of nilpotent groups of Hirsch length six

1995 ◽  
Vol 117 (3) ◽  
pp. 431-438 ◽  
Author(s):  
Charles Cassidy ◽  
Caroline Lajoie
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Torsion Free

AbstractIn this paper, we characterize the genus of an arbitrary torsion-free finitely generated nilpotent group of class two and of Hirsch length six by means of a finite number of arithmetical invariants. An algorithm which permits the enumeration of all possible genera that can occur under the conditions above is also given.

Localization, Algebraic Loops and H-Spaces I

1979 ◽  
Vol 31 (2) ◽  
pp. 427-435 ◽  
Author(s):  
Albert O. Shar
Keyword(s):  
Nilpotent Group ◽  
Algebraic Structure ◽  
Nilpotent Groups ◽  
Topological Spaces ◽  
Finitely Generated ◽  
Cw Complex ◽  
Topological Localization ◽  
The Relationship

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.

scholarly journals DISTORTION IN FREE NILPOTENT GROUPS

2010 ◽  
Vol 20 (05) ◽  
pp. 661-669 ◽  
Author(s):  
TARA C. DAVIS
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Nilpotent Group

We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.

Sign in / Sign up

Export Citation Format

Share Document