2007 Fiscal Year Final Research Report Summary
Research on properties of generating functions for permutation representations and their applications.
| Project/Area Number |
17540002
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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 | Muroran Institute of Technology |
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Principal Investigator |
TAKEGAHARA Yugen Muroran Institute of Technology, Faculty of Engineering, Professor (10211351)
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| Co-Investigator(Kenkyū-buntansha) |
CHIGIRA Naoki Muroran Institute of Techinology, Faculty of Engineering, Associate Professor (40292073)
YAMAZAKI Noriaki Muroran Institute of Techinology, Faculty of Engineering, Associate Professor (90333658)
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| Project Period (FY) |
2005 – 2007
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| Keywords | finitely generated group / profinite group / exponential generating function / nilpotent group / finite abelian group / free product / finite group / Burnside ring |
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| Research Abstract |
a) For each natural number n, let G_n be the kernels of certain linear characters of the wreath product of a finite group by the symmetric group S_n on n letters. When A is a finitely generated group or a profinite group, the exponential generating function for the number of homomorphisms from A to G_n is determined.
b) For a finite group G, the properties of the function T_n from G to the nonnegative integers such that for each g, T_n (g) is the number of sequences (x_1,x_2,…,x_n) of elements of G satisfying the higher commutator [x_1,x_2,…,x_n]=g are obtained.
c) It is obtained that a finite group G is nilpotent of class n if and only if a certain matrix determined from the character table of G is nilpotent of index n.
d) A certain congruence equation modulo p for the number of subgroups of index d, d a fixed natural number, in the free product of finite abelian groups is obtained.
e) Suppose that a finite group A is an operator group of a finite group G, and consider G as a right A-set. A free right G-set Y is called (A,G)-set if Y is a left A-set with the action given by a(yg)=(ay)^ag, a∈A, y∈Y, g∈G. An (A,G)-set is called simple if it is a transitive A-set. A complete set of representatives of isomorphism classes of simple (A,G)-sets is determined. The Grothendieck ring of the category of the (A,G)-set is defined, and some properties of the Burnside ring of a finite group are generalized. Moreover, an exlicit formula of Brauer's induction theorem is obtained as an application of the theory.
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Research Products
(13 results)
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[Presentation] べき零群と指標表2006
Author(s)
竹ヶ原裕元
Organizer
RIMS研究集会、群論とその周辺
Place of Presentation
京都大学数理解析研究所
Year and Date
2006-12-18
Description
「研究成果報告書概要(和文)」より
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