On the Structure of the Mislin Genus of a Pullback

Abstract

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set G(Q) if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations Qp and Rp are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback Ht from the l-equivalences Hi→H and Hj→H, t≡(i+j)mods, where s=|G(H)|, and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product G×K of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial.