On a family of Serre classes of nilpotent groups (Q687578)

Let \(N\) be a \(P\)-local nilpotent group. \(N\) is called finitely generated (fg) as \(P\)-local group, if there is a finite subset \(S\) of \(N\) such that the smallest \(P\)-local subgroup of \(N\) containing \(S\) is \(N\) itself. A nilpotent group \(G\) is finitely generated at every prime \(p\) (fg\(p\)) if \(G_ p\) is fg as \(p\)-local group for all primes \(p\). The author and \textit{J. Roitberg} [Houston J. Math. 2, 525-559 (1976; Zbl 0342.55009)] extended the notion of a Serre class of abelian groups to nilpotent groups as follows: A non-empty class of nilpotent groups \(\mathcal N\) is a Serre class if \(\mathcal N\) satisfies the following properties: Given a central extension \(N \rightarrowtail G \twoheadrightarrow Q\) of nilpotent groups, we have \(G\in {\mathcal N} \Leftrightarrow N\), \(Q \in {\mathcal N}\); If \(A\), \(B\) are abelian and \(A,B \in {\mathcal N}\), then \(A\otimes B\), \(\text{Tor}(A,B)\), \(H_ kA \in {\mathcal N}\) where \(k\geq 1\). The author investigates the fg as \(P\)-local and fg\(p\) nilpotent groups. It is proved that such groups form Serre classes.