### 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 |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*319*(9), 3567-3587. https://doi.org/10.1016/j.jalgebra.2007.08.033