Research on relative difference sets of affine type
| Project/Area Number |
17540034
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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 | Kumamoto University |
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Principal Investigator |
HIRAMINE Yutaka Kumamoto University, Faculty of Education, Professor (30116173)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Atsumi Kumamoto University, Faculty of Science, Professor (90040120)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥2,540,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | relative difference set / group / group ring / character / affine type / design / transversal design / factor set / multiplier / finite field / p-group / meta-cyclic group / semiregular type / cyclotomic field / Relative difference set / simple group / dihedral group |
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| Research Abstract |
We study relative difference sets in the simple group Alt (5) and show that up to equivalence there exist exactly two nontrivial relative difference sets in Alt (5), which are of affine type. It is conjectured that there exists no nontrivial ordinary difference set in dihedral groups. In this connection we also study relative difference sets in dihedral groups and show that there is no nontrivial relative difference set of semiregular type or of affine type in them. Concerning semiregular relative difference sets S. L. Ma and B. Schmidt conjectured that the only semiregular relative difference set in Zp^2 × Zp^2 is a trivial one. Using characters of Zp^2 × Zp^2 we prove that the conjecture is true when p>3. Therefore it is conceivable that there exists no nontrivial semiregular relative difference set in Zp^n ×Zp^n for any n>1. Moreover, we study relation between affine difference sets and group extensions. Let D be an abelian affine difference set of odd order n in a group G of order n2-l with respect to a subgroup N of order n-1. Set H=G/N={t o, t n). Then we can select a suitable factor set f: H×H N such that g (σ)=&.σ, to)f (σ, t i)-f (σ, tn) satisfies (1) g (σm)=g (σ)m for any divisor m of n and (2) g (σ)=g (σ1) for any a in H. Using these nice properties we prove that the multiplicative order of m mod exp (N) is at most 2 times the multiplicative order of m mod exp (H). As an application we know that n is a power of a prime if n ≦100000 with 155 exceptions
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Report
(4 results)
Research Products
(43 results)
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[Presentation] アフィン差集合と乗数2006
Author(s)
平峰 豊
Organizer
デザイン理論とその周辺
Place of Presentation
山形県上山市
Year and Date
2006-11-21
Description
「研究成果報告書概要(和文)」より
Related Report
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