Almost Completely Decomposable Torsion Free Abelian Groups

Abstract

A finite rank torsion free abelian group $G$ is almost completely decomposable if there exists a completely decomposable subgroup $C$ with finite index in $G$. The minimum of $[G:C]$ over all completely decomposable subgroups $C$ of $G$ is denoted by $i(G)$. An almost completely decomposable group $G$ has, up to isomorphism, only finitely many summands. If $i(G)$ is a prime power, then the rank 1 summands in any decomposition of $G$ as a direct sum of indecomposable groups are uniquely determined. If $G$ and $H$ are almost completely decomposable groups, then the following statements are equivalent: (i) $G \oplus L \approx H \oplus L$ for some finite rank torsion free abelian group $L$. (ii) $i(G) = i(H)$ and $H$ contains a subgroup $G’$ isomorphic to $G$ such that $[H:G’]$ is finite and prime to $i(G)$. (iii) $G \oplus L \approx H \oplus L$ where $L$ is isomorphic to a completely decomposable subgroup with finite index in $G$.