2004 Fiscal Year Final Research Report Summary
A study of representations and finite group actions realizing a given fixed point set
| Project/Area Number |
14540084
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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 |
Geometry
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| Research Institution | KYUSHU UNIVERSITY (2004)
九州芸術工科大学 (2002-2003) |
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Principal Investigator |
SUMI Toshio Kyushu University, Faculty of Design, Associate Professor, 大学院・芸術工学研究院, 助教授 (50258513)
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| Co-Investigator(Kenkyū-buntansha) |
IWASE Norio Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (60213287)
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| Project Period (FY) |
2002 – 2004
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| Keywords | gap modules / gap group / finite group actions |
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| Research Abstract |
Let G be a finite group not of prime power order. For a prime p, we denote by O^p(G), called the Dress group of type p, the smallest normal subgroup of G with p-power index. The group G is said to be a gap group if there exists a G-module V such that dim V^<O^2(G)>=0 for any prime p and dim V^P>2dim V^H for any pair (P,H) of subgroups of G with some condition. Note that a Dress subgroup of a gap group G is not of prime power order.
I show that G is a gap group if and only if all subgroups L of G, possessing cyclic quotients L/O^2(G), are gap groups. Now assume that G/O^2(G) is cyclic. As viewing the series of normal groups
G=G_0〓G_1〓G_2〓【triple bond】〓G_k=O^2(G),[G_j,G_<j-1>]=2,
I obtained that G is a gap group if and only if each G_j(0<j<k) is a gap group. I define a subset E_j of 2-elements h of G_j＼G_<j-1> by using a form of the centralizer C_G(h). We can easily decide whether the set E_j is empty or not, for example, letting j>1,E_j is not empty if there exists an element of G_j＼G_<j-1> not of prime power order. It is a little bit complication to decide whether E_1 is empty. Then I showed that all E_j are nonempty if and only if G is a gap group. Furthermore, I obtained that if G×【triple bond】×G is a gap group,then so is G×G.
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