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 |
|---|---|
| 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
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 journal › Article
}
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 -