On Special Group-Automorphisms and Their Composition

Abstract

Let G be a group and φ an automorphism of G. We say that φ is a pseudo-identity (pi) if, for each x ∈ G, there exists a finitely generated (fg) subgroup K = Kx(φ) of G such that x ∈ K and φ|K is an automorphism of K. It has been shown [1, 3] that such special automorphisms of abelian or nilpotent groups play an important role in homotopy theory; and it was indicated in [2] that their purely algebraic properties might well repay study.The following facts about pi's are elementary.PROPOSITION 0.1. Let φ be an automorphism of G and let n be a non-zerointeger. Then φ is pi if and only if φn is pi.PROPOSITION 0.2. Let φ be a pseudo-identity of G and a an automorphismof G. Then αφα-1 is a pseudo-identity.