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

Katsuya Eda, Vlasta Matijević

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)3567-3587
    Number of pages21
    JournalJournal of Algebra
    Volume319
    Issue number9
    DOIs
    Publication statusPublished - 2008 May 1

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    Torsion-free Abelian Group
    Subgroup
    Distinct
    Direct Limit
    Free Group
    Natural number
    Abelian group
    Classify
    Integer

    Keywords

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

    ASJC Scopus subject areas

    • Algebra and Number Theory

    Cite this

    Finite index supergroups and subgroups of torsionfree abelian groups of rank two. / Eda, Katsuya; Matijević, Vlasta.

    In: Journal of Algebra, Vol. 319, No. 9, 01.05.2008, p. 3567-3587.

    Research output: Contribution to journalArticle

    Eda, Katsuya ; Matijević, Vlasta. / Finite index supergroups and subgroups of torsionfree abelian groups of rank two. In: Journal of Algebra. 2008 ; Vol. 319, No. 9. pp. 3567-3587.
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