Localization and I A-automorphisms of finitely generated, metabelian, and torsion-free nilpotent groups

Abstract

Given a nilpotent group G and a prime p, there is a unique p-local group G(p) which is, in some sense, the “best approximation” to G among all p-local nilpotent groups. G(p) is called the p-localization of G. Let IA(G) be the group of automorphisms of G that induce the identity on G/[G, G]. IA(G) turns out to be nilpotent so its p-localization exists. Two groups are said to be in the same localization genus if their p-localizations are isomorphic for all p. We prove that if two finitely generated, torsion-free nilpotent, and metabelian groups lie in the same localization genus, their IA-groups also lie in the same localization genus. The method of proof involves basic sequences and commutator calculus.