Amalgams of torsion free nilpotent groups
A group \(C\) is called amalgam of \(A\) and \(B\) over \(D\) in this paper, if \(A\) and \(B\) are contained in \(C\) and \(D\) is contained in \(A\cap B\); if \(D=A\cap B\), \(C\) is a strict amalgam. Main statement of this article is the existence of a torsion free nilpotent amalgam of two torsionfree nilpotent groups \(A\) and \(B\) provided they have central series which coincide when intersected with \(D\). By imposing another ``interlocking'' condition on the central series it is possible (in contrast to the torsion case, where the corresponding first statement is known) to bound the nilpotency class of the amalgam. The corresponding results for rational nilpotent Lie algebras are also deduced. Using the embedding (indicated by Hirsch) of a finitely generated nilpotent group into the direct product of a torsion free and a finite nilpotent group and proceeding with the finite parts according to Higman it is possible to construct also nilpotent amalgams of finitely generated nilpotent groups. The more general definition of amalgam is needed for the torsion free case. - The actual construction of the amalgam is first done for divisible groups and follows an inductive building up method not to be reproduced here in detail.
scientific article; zbMATH DE number 3939574
scientific article