scholarly journals Some Notes on Relative Commutators

2020 ◽  
Vol 2 (2) ◽  
pp. 65-70
Author(s):  
Masoumeh Ganjali ◽  
Ahmad Erfanian
Keyword(s):  
Nilpotent Group ◽  
Cyclic Group ◽  
Commutator Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Perfect Group ◽  
Group A ◽  
Even Order ◽  
Definition Of

Let G be a group and α ϵ Aut(G).  An α-commutator of elements x, y ϵ G is defined as [x, y]α = x-1y-1xyα. In 2015, Barzegar et al. introduced an α-commutator of elements of G and defined a new generalization of nilpotent groups by using the definition of α-commutators which is called an α-nilpotent group. They also introduced an α-commutator subgroup of G, denoted by Dα(G) which is a subgroup generated by all α-commutators. In 2016, an α-perfect group, a group that is equal to its α-commutator subgroup, was introduced by authors of this paper and the properties of such group was investigated. They proved some results on α-perfect abelian groups and showed that a cyclic group G of even order is not α-perfect for any α ϵ Aut(G). In this paper, we may continue our investigation on α-perfect groups and in addition to studying the relative perfectness of some classes of finite p-groups, we provide an example of a non-abelian α-perfect 2-group.

scholarly journals On almost finitely generated nilpotent groups

1996 ◽  
Vol 19 (3) ◽  
pp. 539-544 ◽  
Author(s):  
Peter Hilton ◽  
Robert Militello
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Local Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Finitely Generated

A nilpotent groupGis fgp ifGp, is finitely generated (fg) as ap-local group for all primesp; it is fg-like if there exists a nilpotent fg groupHsuch thatGp≃Hpfor all primesp. The fgp nilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

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.

scholarly journals Finite nilpotent groups that coincide with their 2-closures in all of their faithful permutation representations

2018 ◽  
Vol 17 (04) ◽  
pp. 1850065
Author(s):  
Alireza Abdollahi ◽  
Majid Arezoomand
Keyword(s):  
Nilpotent Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
Finite Nilpotent Group ◽  
Closed Group ◽  
Odd Order ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

Let [Formula: see text] be any group and [Formula: see text] be a subgroup of [Formula: see text] for some set [Formula: see text]. The [Formula: see text]-closure of [Formula: see text] on [Formula: see text], denoted by [Formula: see text], is by definition, [Formula: see text] The group [Formula: see text] is called [Formula: see text]-closed on [Formula: see text] if [Formula: see text]. We say that a group [Formula: see text] is a totally[Formula: see text]-closed group if [Formula: see text] for any set [Formula: see text] such that [Formula: see text]. Here we show that the center of any finite totally 2-closed group is cyclic and a finite nilpotent group is totally 2-closed if and only if it is cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.

scholarly journals Comments on Y. O. Hamidoune's Paper ‘Adding Distinct Congruence Classes’

2016 ◽  
Vol 25 (5) ◽  
pp. 641-644
Author(s):  
BÉLA BAJNOK
Keyword(s):  
Cyclic Group ◽  
Abelian Groups ◽  
Short Note ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Even Order ◽  
Congruence Classes

The main result in Y. O. Hamidoune's paper ‘Adding distinct congruence classes' (Combin. Probab. Comput.7 (1998) 81–87) is as follows. If S is a generating subset of a cyclic group G such that 0 ∉ S and |S| ⩾ 5, then the number of sums of the subsets of S is at least min(|G|, 2|S|). Unfortunately, the argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result is actually true for all finite abelian groups.

scholarly journals Diameters of random Cayley graphs of finite nilpotent groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniel El-Baz ◽  
Carlo Pagano
Keyword(s):  
Nilpotent Group ◽  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Limiting Distribution ◽  
General Inequality ◽  
Bounded Rank

Abstract We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

Commutator length of abelian-by-nilpotent groups

1998 ◽  
Vol 40 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Mehri Akhavan-Malayeri ◽  
Akbar Rhemtulla
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Nilpotent Groups ◽  
Free Nilpotent Group ◽  
Commutator Length

AbstractLet G be a group and C = [G, G] be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. The exact values of c{G) are computed when G is a free nilpotent group or a free abelian-by-nilpotent group. If G is a free nilpotent group of rank n>2 and class c>2, c(G) = [n/2] if c = 2 and c(G) = n if c>2. If G is a free abelian-by-nilpotent group of rank n > 2 then c(G) = n.

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 A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

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.

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