RESEARCH OF PURIFIABLE AND QUASI-PURIFIABLE SUBGROUPS
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants|
|Research Institution||TOBA NATIONAL COLLEGE OF MARITIME TECHNOLOGY|
OKUYAMA Takashi TOBA NATIONAL COLLEGE OF MARITIME TECHNOLOGY ASSOCIATE PROFESSOR, 助教授 (20177190)
SANAMI Manabu TOBA NATIONAL COLLEGE OF MARITIME TECHNOLOGY LECTURER, 講師 (10226029)
NASHIRO Hiroaki TOBA NATIONAL COLLEGE OF MARITIME TECHNOLOGY PROFESSOR, 教授 (40043252)
|Project Period (FY)
2001 – 2002
Completed(Fiscal Year 2002)
|Budget Amount *help
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 2002 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2001 : ¥800,000 (Direct Cost : ¥800,000)
|Keywords||Purifiable Subgroups / Quasi-Purifiable Subgroups / p-overhang set / The maximal torsion subgroup / Torsion-Free subgroup / T-High Subgroup / Height-Matrices / Torsion-Free Rank / 準純粋包をもつ部分群 / 純粋包 / p-overhang set / 準純粋包 / empty-p-overhang set部分群 / 最大ねじれ部分群|
In an arbitrary abelian group G, a subgroup A of G is said to be purifiable in G if there exists a pure subgroup H of G containing A which is minimal among the pure subgroups of G that contain A. Such a subgroup H is called a pure hull of A. In general, not all subgroups are purifiable. Now we can pose the following problem: Which subgroup is purifiable in a given group?
We started this project with the problem. In this project, we considered only torsion-free subgroups. Finally, we obtained the following results.
(1) We characterized torsion-free finite rank purifiable subgroups in give groups.
(2) We proved that all pure hulls of purifiable torsion-free subgroups are isomorphic.
We used the result (1) to study the splitting problem. The splitting problem is to characterize mixed groups which are a direct sum of the maximal torsion and torsion-free subgroup. We obtained a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.
A subgroup A of an a
belian group G is said to be quasi-purifiable in G if there exists a pure subgroup K of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup K is called a quasi-pure hull of A in G. We can also pose the following problem.
Which subgroup is quasi-purifiable in a given group?
In this project, for the above problem, we characterized torsion-free rank-one quasi-purifiable subgroups in given groups. We used this result to show how to calculate the height-matrices of straight elements. Height-matrices are important. For example, it is well-known that countable mixed groups H and K of torsion-free rank 1 are isomorphic if and only if T(H) is isomorphic to T(K) and the height-matrices of H and K are equivalent.
If a subgroup A of a group G is quasi-purifiable in G, then there exists a maximal quasi-pure hull of A in G. So we might use the concept of maximal quasi-pure hulls to study the groups whose maximal torsion-subgroups are torsion-complete. These groups whose maximal torsion subgroups are torsion-complete include direct products of cyclic p-groups. This study could be next. Less
Research Products (22results)