On almost finitely generated nilpotent groups (Q1914778)

A nilpotent group \(G\) is finitely generated at every prime (fgp) if \(G_p=G/G^p\) is finitely generated (fg) as \(p\)-local group for all primes \(p\); it is fg-like if there exists a nilpotent group \(H\) such that \(G_p\cong H_p\) for all primes \(p\). The authors prove that for any short exact sequence of nilpotent groups \(G'\to G\to G''\), \(G'\) and \(G''\) are fgp if and only if \(G\) is fgp. For abelian groups it is shown that a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group if fg-like. An analogous result is not valid for nilpotent groups. The authors construct a torsion free nilpotent fg-like group \(H\) of rank 2, admitting a subgroup \(G\), such that (i) \(G\) is not fg-like and (ii) \(G\) is extension of an fg-like abelian group by an fg-like abelian group. Also they study fg-like groups with finite commutator subgroup.