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

Katsuya Eda, Vlasta Matijević

    Research output: Contribution to journalArticle

    2 Citations (Scopus)


    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 languageEnglish
    Pages (from-to)3567-3587
    Number of pages21
    JournalJournal of Algebra
    Issue number9
    Publication statusPublished - 2008 May 1



    • Finite index
    • Rank two
    • Subgroup
    • Supergroup
    • Torsionfree abelian group

    ASJC Scopus subject areas

    • Algebra and Number Theory

    Cite this