Fully inert subgroups of free Abelian groups

Abstract

A subgroup $H$ of an Abelian group $G$ is called fully inert if $(\phi H + H)/H$ is finite for every $\phi \in \mathrm{End}(G)$. Fully inert subgroups of free Abelian groups are characterized. It is proved that $H$ is fully inert in the free group $G$ if and only if it is commensurable with $n G$ for some $n \ge 0$, that is, $(H + nG)/H$ and $(H + nG)/nG$ are both finite. From this fact we derive a more structural characterization of fully inert subgroups $H$ of free groups $G$, in terms of the Ulm–Kaplansky invariants of $G/H$ and the Hill–Megibben invariants of the exact sequence $0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0$.