A nilpotency condition for finitely generated soluble groups

Costantino Delizia

Displaying similar documents to “A nilpotency condition for finitely generated soluble groups”

Infinite locally soluble $k$ -Engel groups

Lucia Serena Spiezia (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we deal with the class $E_{k}^{*}$ of groups $G$ for which whenever we choose two infinite subsets $X$ , $Y$ there exist two elements $x \in X$ , $y \in Y$ such that $[x, \underset{k}{\underset{︸}{y, \dots, y}}] = 1$ . We prove that an infinite finitely generated soluble group in the class $E_{k}^{*}$ is in the class $E_{k}$ of $k$ -Engel groups. Furthermore, with $k = 2$ , we show that if $G \in E_{2}^{*}$ is infinite locally soluble or hyperabelian group then $G \in E_{2}$ .

On absolutely-nilpotent of class $k$ groups

Patrizia Longobardi, Trueman MacHenry, Mercede Maj, James Wiegold (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A group $G$ in a variety $V$ is said to be absolutely- $V$ , and we write $G \in A V$ , if central extensions by $G$ are again in $V$ . Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class $A N_{k}$ of absolutely-nilpotent of class $k$ groups. We prove some closure properties of the class $A N_{k}$ and we show that every nilpotent of class $k$ group can be embedded in an $A N_{k}$ -gvoup. We describe all metacyclic $A N_{k}$ -groups and we characterize $2$ -generator and infinite $3$ -generator $A N_{2}$ -groups....

Groups whose proper subgroups are locally finite-by-nilpotent

Amel Dilmi (2007)

Annales mathématiques Blaise Pascal

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If $𝒳$ is a class of groups, then a group $G$ is said to be minimal non $𝒳$ -group if all its proper subgroups are in the class $𝒳$ , but $G$ itself is not an $𝒳$ -group. The main result of this note is that if $c > 0$ is an integer and if $G$ is a minimal non $(ℒℱ) 𝒩$ (respectively, $(ℒℱ) 𝒩_{c}$ )-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G / F r a t (G)$ is an infinite simple group; where $𝒩$ (respectively, $𝒩_{c}$ , $ℒℱ$ ) denotes the class of nilpotent (respectively, nilpotent of class...

Hurewicz-Serre theorem in extension theory

M. Cencelj, J. Dydak, A. Mitra, A. Vavpetič (2008)

Fundamenta Mathematicae

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The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that $X τ K (H_{k} (L), k)$ for all k ≥ 1, then $X τ K (π_{k} (F), k)$ and $X τ K (π_{k} (L), k)$ for all k ≥ $.$ Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈...

Parallelepipeds, nilpotent groups and Gowers norms

Bernard Host, Bryna Kra (2008)

Bulletin de la Société Mathématique de France

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In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions $2$ and $3$ and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.

The Hughes subgroup

Robert Bryce (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $G$ be a group and $p$ a prime. The subgroup generated by the elements of order different from $p$ is called the Hughes subgroup for exponent $p$ . Hughes [3] made the following conjecture: if $H_{p} (G)$ is non-trivial, its index in $G$ is at most $p$ . There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case $p = 3$ and $G$ arbitrary, and that of Hogan and Kappe [2] concerning the case when $G$ is metabelian, and $p$ arbitrary. A common...

Displaying similar documents to “A nilpotency condition for finitely generated soluble groups”

Infinite locally soluble k -Engel groups

On absolutely-nilpotent of class k groups

Groups whose proper subgroups are locally finite-by-nilpotent

Hurewicz-Serre theorem in extension theory

Parallelepipeds, nilpotent groups and Gowers norms

The Hughes subgroup

Infinite locally soluble $k$ -Engel groups

On absolutely-nilpotent of class $k$ groups