Let $A \subseteq B \subseteq C$ be a chain of finitely generated abelian groups. Is it always possible to find compatible bases for these three groups?
By this, I mean $\mathbf{Z}$-bases $\{a_i\}$ of $A$, $\{b_j\}$ of $B$, and $\{c_k\}$ of $C$ such that each $b_j$ is an integer multiple of a single $c_k$ and similarly each $a_i$ is an integer multiple of a single $b_j$.