On absolutely-nilpotent of class \(k\) groups
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 \(H/M\simeq G\).
infinite 3-generator \(A{\mathcal N}_ 2\)-groups
scientific article; zbMATH DE number 915281