On nilpotent groups which are finitely generated at every prime (Q1803505)

Let \(N\) be a \(P\)-local nilpotent group, where \(P\) is a family of primes. A collection of elements \(\{x_ i\}\) in \(N\) generates the \(P\)-local subgroup \(M\) of \(N\) if \(M\) is the smallest \(P\)-local subgroup of \(N\) containing the elements \(x_ i\). A \(P\)-local subgroup \(N\) is finitely generated (fg) as a \(P\)-local group if there exists a finite set \(\{x_ 1,\dots,x_ n\}\) of elements of \(N\) which generates it in the above sense. A nilpotent group \(G\) is finitely generated at every prime (fg\(p\)) if \(G_ p\) is a finitely generated \(p\)-local group for all primes \(p\). In this work some properties which fg\(p\) nilpotent groups have in common with fg nilpotent groups are discovered.