Genus of a pullback

Abstract

Let Q be a finitely generated nilpotent group. The Mislin genus G (Q) is defined to be the set of isomorphism classes of finitely generated nilpotent groups R such that for every prime p, the p-localizations Rp and Qp are isomorphic. If the group Q has a finite commutator sub group, the genus admits an abelian group structure. In this work, we study the Mislin genus of nilpotent groups through pullbacks. For a relatively prime pair of natural numbers (n, u), let Q = hx, y | x n = 1, yxy−1 = x u i, where Q is a pullback. We compute the genus G (Q) and we investigate the relationship between the genera of these nilpo tent groups.