2000 Fiscal Year Final Research Report Summary
Research on the relation between relative difference sets in non-abelian groups and group characters
| Project/Area Number |
11640037
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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 |
YUTAKA Hiramine Kumamoto Univ., Faculty of Education, Professor, 教育学部, 教授 (30116173)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Atsumi Kumamoto Univ., Faculty of Science, Assistant Professor, 理学部, 助教授 (90040120)
ITOH Jin-ichi Kumamoto Univ., Faculty of Education, Assistant Professor, 教育学部, 助教授 (20193493)
KANEMARU Tadayoshi Kumamoto Univ., Faculty of Education, Professor, 教育学部, 教授 (30040033)
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| Project Period (FY) |
1999 – 2000
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| Keywords | difference set / group / character / design / group ring |
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| Research Abstract |
In this research we studied the relation between semi-regular relative difference sets (RDS) and group characters. Our main results are as follows :
Theorem 1. Let G be a group with an abelian normal subgroup A of index 2 inverted by an element t ∈ G＼A of order 2. Assume that G has a semi-regular RDS, say R, relative to a normal subgroup U of G.Then a Sylow p-subgroup of A is non-cyclic for any prime p dividing u.
J.A.Davis has constructed examples of semi-regular RDS in Q_<2^m>. In general we obtained the following.
Theorem 2. Let R be a semi-regular RDS in a dicyclic group Q_<4m> relative to a normal subgroup U.If |U| is even, then |U|=2.
Concerning dihedral groups D_<2m> and semi-dihedral groups SD_<2^m>, we have the following result.
Theorem 3 There is no semi-regular RDS in any dihedral group.
As a conclusion in this research we have shown the following.
Theorem 4 Let G be a non-abelian p-group with a maximal cyclic subgroup. If G contains a semi-regular RDS relative to a normal subgroup U, then one of the following holds :
(i) G【similar or equal】Q_<2^n> and U【similar or equal】Z_2,
(ii) G【similar or equal】M_n (p) and U【similar or equal】Z_p with p an odd prime,
(iii) G【similar or equal】M_4 (2) and U【similar or equal】Z_2×Z_2.
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