The group of inertial automorphisms of an abelian group (0)

We study the group $IAut(A)$ generated by inertial automorphisms of an abelian group $A$, that is automorphisms $\g$ with the property $|\la X,X\g\ra/X|<\infty$ for each $X\le A$. Clearly $IAut(A)$ contains the group 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. Here we have that $IAut(A)$ is abelian-by-(locally finite) in the case $A$ is periodic, while in the general case it is not even (locally nilpotent)-by-(locally finite). However $IAut(A)$ has a normal subgroup $\G$ such that $IAut(A)/\G$ is locally finite and $\G$ acts by means of power automorphisms on its derived subgroup, which is abelian. Moreover we describe into details the structure of $IAut(A)$ in some relevant cases for $A$.\\ \phantom{xxx} We apply our techniques also to the study of groups whose subnormal subgroups are inert.