Finite index supergroups and subgroups of torsion free abelian groups of rank two

Every torsionfree abelian group A of rank two is a subgroup of Q circle plus Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2 x 2-matrices as the bonding maps. Using these direct systems we classify all subgroups of Q circle plus 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 circle plus Q and all supergroups are subgroups of Q circle plus Q: (1) for each natural numbers there are Sigma(q vertical bar s),(gcd(p,q)=1) q s-index supergroups and also Sigma(q vertical bar s),(gcd(p,q)=1) s-index subgroups; (2) each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index, subgroups are non-isonnorphic. (c) 2008 Elsevier Inc. All rights reserved.