On the Structure of the Mislin Genus of a Pullback

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 (Formula presented.) if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations (Formula presented.) and (Formula presented.) 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 (Formula presented.) from the l-equivalences (Formula presented.) and (Formula presented.), (Formula presented.), where (Formula presented.), and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product (Formula presented.) of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial.