Finite index supergroups and subgroups of torsionfree abelian groups of rank two

Every torsionfree abelian group A of rank two is a subgroup of Q⊕Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2×2-matrices as the bonding maps. Using these direct systems we classify all subgroups of Q⊕Q which are finite index supergroups of A or finite index subgroups of A. Using this classification we prove that for each prime p there exists a torsionfree abelian group A satisfying the following, where A⩽Q⊕Q and all supergroups are subgroups of Q⊕Q:(1)for each natural number s there are s-index supergroups and also s-index subgroups;(2)each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index subgroups are non-isomorphic.