ON GENUS AND EMBEDDINGS OF TORSION-FREE NILPOTENT GROUPS OF CLASS-2

We study embeddings between torsion-free nilpotent groups having isomo rphic localizations. Firstly, we show that for finitely generated tors ion-free nilpotent groups of nilpotency class 2, thee property of havi ng isomorphic P-localizations (where P denotes any set of primes) is e quivalent to the existence of mutual embeddings of finite index not di visible by ally prime in P. We then focus on a certain family Gamma of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multi plication of equivalence classes of groups in Gamma induced by the ide al class group structure can be described by means of certain pull-bac k diagrams reflecting the existence of enough embeddings between membe rs of each Mislin genus. In this sense, the family Gamma resembles the family N-0 of infinite, finitely generated nilpotent groups with fini te commutator subgroup. We also show that, in further analogy with N-0 , two groups in Gamma with isomorphic localizations at every prime hav e isomorphic localizations at every finite set of primes. We supply co unterexamples showing that this is not true in general, neither for fi nitely generated torsion-free nilpotent groups of class 2 nor for tors ion-free abelian groups of finite rank.