Invariants for a Class of Torsion-Free Abelian Groups
D. Arnold, C. Vinsonhaler · Proceedings of the American Mathematical Society · 1989
In this note we present a complete set of quasi-isomorphism invariants for strongly indecomposable abelian groups of the form $G = G({A_1}, \ldots ,{A_n})$. Here ${A_1}, \ldots ,{A_n}$ are subgroups of the rationals $Q$ and $G$ is the kernel of $f:{A_1} \oplus \cdots \oplus {A_n} \to Q$, where $f({a_1}, \ldots ,{a_n}) = \Sigma {a_i}$. The invariants are the collection of numbers ${\text {rank}} \cap \{ G[\sigma ]|\sigma \in M\}$, where $M$ ranges over all subsets of the type lattice generated by $\left \{ {{\text {type}}({A_i})} \right \}$. Our results generalize the classical result of Baer for finite rank completely decomposable groups, as well as a result of F. Richman on a subset of the groups of the form $G({A_1}, \ldots ,{A_n})$.