On the fixed-point set of an automorphism of a group

Abstract

Let O be an automorphism of a group G. Under variousfiniteness or solubility hypotheses, for example under polycyclicity, the commutator subgroup [G; O] has finite index in G if thefixed-point set CG(O) of O in G isfinite, but not conversely, even for polycyclic groups G. Here we consider a stronger, yet natural, notion of what it means for [G;O] to have finite index' in G and show that in many situations, including G polycyclic, it is equivalent to CG(O) being finite.