### 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

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

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 319, no. 9, pp. 3567-3587. https://doi.org/10.1016/j.jalgebra.2007.08.033

}

TY - JOUR

T1 - Finite index supergroups and subgroups of torsionfree abelian groups of rank two

AU - Eda, Katsuya

AU - Matijević, Vlasta

PY - 2008/5/1

Y1 - 2008/5/1

N2 - 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.

AB - 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.

KW - Finite index

KW - Rank two

KW - Subgroup

KW - Supergroup

KW - Torsionfree abelian group

UR - http://www.scopus.com/inward/record.url?scp=41149115989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41149115989&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2007.08.033

DO - 10.1016/j.jalgebra.2007.08.033

M3 - Article

VL - 319

SP - 3567

EP - 3587

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 9

ER -