The group of inertial automorphisms of an abelian group

Abstract

We study the group $IAut(A)$ generated by the inertial automorphisms of an abelian group $A$, that is, automorphisms $\gamma$ with the property that each subgroup $H$ of $A$ has finite index in the subgroup generated by $H$ and $H\gamma$. 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). In particular, $IAut(A)$ has a normal subgroup $\Gamma$ such that $IAut(A)/\Gamma$ is locally finite and ${\Gamma }^{\prime }$ is an abelian periodic subgroup whose all subgroups are normal in $\Gamma$. In the case when $A$ is periodic, $IAut(A)$ results 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 $IAut(A)$ in some other cases for $A$.

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References in corpus (1)

• On groups whose subnormal subgroups are inert

Cited by in corpus (2)

• On groups whose subnormal subgroups are inert
• On the ring of inertial endomorphisms of an abelian group