Strong genus of nilpotent groups of Hirsch length six

Abstract

We give a characterization in terms of finitely many arithmetical invariants for two finitely generated, torsion-free, nilpotent groups of nilpotency class two and Hirsch length six to have isomorphic localizations at every finite set of primes. This result is obtained through adequate reduction of an arbitrary binary integral quadratic form. We derive some consequences and, in particular, we characterize completely those groups N in the above class of groups, for which the Mislin genus coincides with the strong genus (i.e. those groups N for which, whenever a nilpotent group M satisfies M_p ≅ N_p for every prime p, then M_P ≅ N_P for every finite set of primes P).