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.