If two finitely generated, torsion-free, nilpotent groups of class
two satisfy the two-arrow property
(that is, they embed into each other with finite, relatively prime indices),
then they necessarily belong to
the same Mislin genus (that is, they have isomorphic localizations at every
prime). Here we show that
the other implication is false in general. We even provide counterexamples
in the case where both groups
have isomorphic localizations at every finite set of primes of bounded
cardinality. The latter equivalence
relation leads us to introduce the notion of n-genus for every
positive integer n, which we show to be
meaningful in various contexts. In particular, the two-arrow property is
related to the n-genus in the
context of topological spaces.