Generalized Rigid Metabelian Groups

We study the generalized rigid groups ( r -groups), in the metabelian case in more detail. The periodic r -groups are described. We prove that each divisible metabelian r -group decomposes as a semidirect product of two abelian subgroups, each metabelian r -group independently embeds into a divisible metabelian r -group, and the intersection of each collection of divisible subgroups of a metabelian r -group is divisible too.