The group of inertial automorphisms of an abelian group

Ulderico Dardano, Silvana Rinauro

Published 2014-03-17, updated 2014-11-06Version 3

We study the group $IAut(A)$ generated by inertial automorphisms of an abelian group $A$, that is automorphisms $\gamma$ with the property $|\langle X,X\gamma\rangle /X|<\infty$ for each $X\le A$. 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. Here we show that $IAut(A)$ is also metabelian-by-(locally finite). In particular, $IAut(A)$ has a normal subgroup $\Gamma$ such that $IAut(A)/\Gamma$ is locally finite and $\Gamma$ acts by means of power automorphisms on its derived subgroup, which is a periodic abelian group. In the case when $A$ is periodic, $IAut(A)$ results to be even abelian-by-(locally finite), while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover we describe into details the structure of $IAut(A)$ in some relevant cases for $A$.