2004 Fiscal Year Final Research Report Summary
ABELIAN GROUPS OF TORSION-FREE RANK 1
| Project/Area Number |
15540052
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | OITA UNIVERSITY (2004)
Toba National College of Maritime Technology (2003) |
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Principal Investigator |
OKUYAMA Takashi OITA UNIVERSITY, FACULTY OF ENGINEERING, PROFESSOR, 工学部, 教授 (20177190)
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| Co-Investigator(Kenkyū-buntansha) |
TANAKA Yasuhiko OITA UNIVERSITY, FACULTY OF ENGINEERING, ASSOCIATE PROFESSOR, 工学部, 助教授 (70244150)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Torsion-complete group / Purifiable subgroup / T-high subgroup / Quasi-purifiable subgroup / Quasi-pure hull / Minimal direct summand / ADE decomposable group / Mixed basic subgroup |
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| Research Abstract |
First we proved that all subgroups of abelian groups G whose maximal torsion subgroups are torsion-complete are quasi-purifiable in G. We can easily see that there exist maximal quasi-pure hulls of quasi-purifiable subgroups by Zom's Lemma. We also proved that all maximal quasi-pure hulls of quasi-purifiable subgroups of the groups G are isomorphic. As an application of this result, we proved that, for every subgroup A of torsion-complete groups G, there exists a minimal direct summand H of G containing A and such a minimal direct summand H is a minimal pure torsion-complete subgroup of G containing A. When we apply the above result to arbitrary abelian group G, we can show that all groups whose maximal torsion subgroups are torsion-complete are ADE decomposable groups. Applying the same result to abelian groups of torsion-fiee rank 1, we can prove that if X and Y are ADE decomposable groups of torsion-free rank 1, then G and H are isomorphic if and only if the maximal torsion subgroups T(X) and T(Y) are isomorphic and the height-matrices H(X) and H(Y) are equivalent.
We could extend the concept of basic subgroups of torsion groups to arbitrary abelian groups. We named the subgroups "mixed basic subgroups". First, we showed that not all mixed basic subgroups of arbitray abelian groups are isomorphic, though all basic subgroups of torsion groups are isomorphic.
Moreover, we proved that there exists a T-high subgroup L of an abelian group G of torsion-free rank 1 such that type(L) is less than or equal to type(A) for every T-high sugroup A of G. We used this result to characterize the abelian group of torsion-free rank 1 all of whose T-hih subgroups are isomorphic.
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