FROM MISLIN GENUS TO STRONG GENUS

Abstract

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.