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 f with the property that each subgroup H of A has finite index in the subgroup generated by H and f(H) . 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 N such that IAut(A) /N is locally finite and the derived subgroup of N 0 is an abelian periodic subgroup all of whose subgroups are normal in N. 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)