Some properties of FC-groups which occur as automorphism groups

Abstract

We prove that if G G is a group such that Aut G G is a countably infinite torsion F C FC -group, then Aut G G contains an infinite locally soluble, normal subgroup and hence a nontrivial abelian normal subgroup. It follows that a countably infinite subdirect product of nontrivial finite groups, of which only finitely many have nontrivial abelian normal subgroups, is not the automorphism group of any group.