On absolutely-nilpotent of class $k$ groups

Longobardi, Patrizia, et al. "On absolutely-nilpotent of class \( k \) groups." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.4 (1995): 201-209. <http://eudml.org/doc/244336>.

@article{Longobardi1995,
abstract = {A group \( G \) in a variety \( \mathfrak\{V\} \) is said to be absolutely-\( \mathfrak\{V\} \), and we write \( G \in A \mathfrak\{V\} \), if central extensions by \( G \) are again in \( \mathfrak\{V\} \). Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class \( A \mathfrak\{N\}\_\{k\} \) of absolutely-nilpotent of class \( k \) groups. We prove some closure properties of the class \( A \mathfrak\{N\}\_\{k\} \) and we show that every nilpotent of class \( k \) group can be embedded in an \( A \mathfrak\{N\}\_\{k\} \)-gvoup. We describe all metacyclic \( A \mathfrak\{N\}\_\{k\} \)-groups and we characterize \( 2 \)-generator and infinite \( 3 \)-generator \( A \mathfrak\{N\}\_\{2\} \)-groups. Finally we study extensions \( 1 \to N \to H \to G \to 1 \), with \( N \le \zeta\_\{n\} (H) \), the \( n \)-centre of $ H $, with\( n > 1 \).},
author = {Longobardi, Patrizia, MacHenry, Trueman, Maj, Mercede, Wiegold, James},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Variety; Central extension; Nilpotent group; varieties of groups; absolutely-nilpotent groups; central extensions; free presentations; lower central series; metacyclic -groups; infinite 3-generator -groups; nilpotent extensions},
language = {eng},
month = {12},
number = {4},
pages = {201-209},
publisher = {Accademia Nazionale dei Lincei},
title = {On absolutely-nilpotent of class \( k \) groups},
url = {http://eudml.org/doc/244336},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Longobardi, Patrizia
AU - MacHenry, Trueman
AU - Maj, Mercede
AU - Wiegold, James
TI - On absolutely-nilpotent of class \( k \) groups
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/12//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 4
SP - 201
EP - 209
AB - A group \( G \) in a variety \( \mathfrak{V} \) is said to be absolutely-\( \mathfrak{V} \), and we write \( G \in A \mathfrak{V} \), if central extensions by \( G \) are again in \( \mathfrak{V} \). Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class \( A \mathfrak{N}_{k} \) of absolutely-nilpotent of class \( k \) groups. We prove some closure properties of the class \( A \mathfrak{N}_{k} \) and we show that every nilpotent of class \( k \) group can be embedded in an \( A \mathfrak{N}_{k} \)-gvoup. We describe all metacyclic \( A \mathfrak{N}_{k} \)-groups and we characterize \( 2 \)-generator and infinite \( 3 \)-generator \( A \mathfrak{N}_{2} \)-groups. Finally we study extensions \( 1 \to N \to H \to G \to 1 \), with \( N \le \zeta_{n} (H) \), the \( n \)-centre of $ H $, with\( n > 1 \).
LA - eng
KW - Variety; Central extension; Nilpotent group; varieties of groups; absolutely-nilpotent groups; central extensions; free presentations; lower central series; metacyclic -groups; infinite 3-generator -groups; nilpotent extensions
UR - http://eudml.org/doc/244336
ER -