On absolutely-nilpotent of class \(k\) groups
type: Item
instance of: scholarly article
title: On absolutely-nilpotent of class \(k\) groups
zbMATH Open document ID: 0862.20028
author: P. Longobardi
author: Trueman MacHenry
author: M. Maj
author: James Wiegold
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1997-05-27T00:00:00Z
full work available at URL: https://eudml.org/doc/244336
review text: This paper studies the class \(A{\mathcal N}_k\) of absolutely-nilpotent groups which is defined to consist of nilpotent groups \(G\) of nilpotency class \(\leq k\) having the property that every central extension by \(G\) is again nilpotent of class \(\leq k\). If \(1\to R\to F\to G\to 1\) is a free presentation of \(G\), then \(G\in A{\mathcal N}_k\) if and only if \(\gamma_{k+1}(F)\subset[F,R]\); this is further equivalent to the condition that \(\inf:H^2(G/\gamma_k(G),T)\to H^2(G,T)\) is an epimorphism, where \(\gamma_k(G)\) is the \(k\)th term in the lower central series of \(G\) and \(T\) is the additive group of rationals \(\text{mod }1\) with trivial action [the reviewer and \textit{L. R. Vermani}, J. Lond. Math. Soc., II. Ser. 6, 129-136 (1972; Zbl 0253.20075)]. Thus absolutely Abelian groups are just those Abelian groups whose Schur multiplicator is trivial [\textit{F. R. Beyl}, J. Pure Appl. Algebra 7, 175-193 (1976; Zbl 0322.18008)]. The authors obtain certain closure properties of the class \(A{\mathcal N}_k\). It is shown, for example, that if \(G\) is a finite \(A{\mathcal N}_k\)-group, then every Cartesian power of \(G\) is in \(A{\mathcal N}_k\). The main results of the paper give a description of the metacyclic \(A{\mathcal N}_k\)-groups, and 2-generator and infinite 3-generator \(A{\mathcal N}_2\)-groups. Passing to higher nilpotent extensions, it is proved that if \(G\) is a 2-generator \(p\)-group of class \(\leq 3\), then, for every integer \(n>1\), there exists a group \(H\) of class \(n+2\) with a normal subgroup \(M\) such that \(M\leq \gamma_n H\) and \(H/M\simeq G\).
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 915281
zbMATH Keywords: varieties of groups
zbMATH Keywords: absolutely-nilpotent groups
zbMATH Keywords: central extensions
zbMATH Keywords: free presentations
zbMATH Keywords: lower central series
zbMATH Keywords: metacyclic \(A{\mathcal N}_ k\)-groups
zbMATH Keywords: infinite 3-generator \(A{\mathcal N}_ 2\)-groups
zbMATH Keywords: nilpotent extensions
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 915281