Quasi-power automorphisms of infinite groups

Abstract

A power automorphism of a group G is an automorphism fixing every subgroup of G. Power automorphisms have been studied by many authors, mainly by C.D.H. Cooper [2]. The set PAutG of all power automorphisms of a group G is a normal, abelian, residually finite subgroup of the full automorphism group AutG of G. The aim of this paper is the study of quasi-power automorphisms of infinite groups. We say that an automorphism of a group G is a quasi-power automorphism if it fixes all but finitely many subgroups of G. It is clear that the set of all quasi-power automorphisms of G is a normal subgroup QAutG of AutG containing PAutG and that QAutG = AutG if G is finite. It is easily verified that quasi-power automorphisms fix all infinite subgroups (see Lemma 2.2 below). Automorphisms fixing infinite subgroups of groups have been studied by M. Curzio, S. Franciosi and F. de Giovanni [3] under the name of I-automorphisms. They prove that, under certain solubility or finiteness conditions for the group G, the group IAutG of all I-automorphisms of G coincides with PAutG, provided G is not a Cernikov group. They also give some sufficient conditions on a non-Cernikov group G to ensure the commutativity of IAutG and exhibit, by contrast, an infinite Cernikov group G such that IAutG is not abelian. Stronger results hold for quasi-power automorphisms. Indeed, if G is an infinite group, then QAutG is always abelian and residually finite, as happens for PAutG. Furthermore, it turns out that the existence of quasi-power automorphisms which are not power automorphisms affects the structure of an infinite group strongly, even if no further condition on this group is imposed. Our main result illustrating this is the following Theorem A, which also gives information on the subgroups which are not fixed under the action of quasi-power automorphisms.