A group of generalized finitary automorphisms of an abelian group

We study the group IAut(A) generated by the inertial automorphisms of an abelian group A, that is, automorphisms g with the property that each subgroup H of A has finite index in the subgroup generated by H and H^g. Clearly, IAut(A) contains the group FAut(A) of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that IAut(A) is (locally finite)-by-abelian. In this paper, we show that IAut(A) is also metabelian-by-(locally finite). More precisely, IAut.A has a normal subgroup G€ such that IAut(A)/G=€ is locally finite and the derived subgroup €G' is an abelian periodic subgroup all of whose subgroups are normal in G€. In the case when A is periodic, IAut(A) turns out to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of IAut(A).