Abstract
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 ∑q | s, gcd (p, q) = 1 q s-index supergroups and also ∑q | s, gcd (p, q) = 1 q 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.
Original language | English |
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Pages (from-to) | 3567-3587 |
Number of pages | 21 |
Journal | Journal of Algebra |
Volume | 319 |
Issue number | 9 |
DOIs | |
Publication status | Published - 2008 May 1 |
Keywords
- Finite index
- Rank two
- Subgroup
- Supergroup
- Torsionfree abelian group
ASJC Scopus subject areas
- Algebra and Number Theory