2004 Fiscal Year Final Research Report Summary
Research on properties of generating functions of the number of special permutation representations and their applications.
| Project/Area Number |
15540002
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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, 工学部, 教授 (10211351)
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| Co-Investigator(Kenkyū-buntansha) |
CHIGIRA Naoki Muroran Institute of Technology, Faculty of Engineering, Associate Professor, 工学部, 助教授 (40292073)
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| Project Period (FY) |
2003 – 2004
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| Keywords | finite group / homomorphism / exponential generation function / symmetric group / complex reflection group / Weyl group / irreducible character / p-adic property |
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| Research Abstract |
a)We study the properties of two p-adic power series exp(f(x)) and exp(g(x)), which appear in the exponential generating function for the number of solutions in the alternating group on n-letters to the equation x^d=1. Moreover, when exp(f(x)) and exp(g(x)) are considered as the exponential generating functions for the sequences {h_n} and {r_n}, respectively, we express h_n and r_n by p-adic analytic functions.
b)Let H be a finite simple group, and identify H with the group consisting of all inner automorphisms of H. Let G be a group of automorphisms of H, and suppose that an element a of G is not an element of H and that a^2 is an element of H. The conjecture : "if a divisor e of the order of H is a multiple of the largest power of 2 dividing the order of H and if the number of solutions in aH to the equation x^<2e>=1 is e, then every element of aH is a solution to the equation x^<2e>=1" is jure if H is the alternating group on n-letters.
c)We consider G_n to be the kernel of a homomorphism from the wreath product of a finite group G with the symmetric group on n-letters which is determined by a homomorphism from G to the cyclic group C_m generated by a primitive m-th root of the unity in the complex numbers. Using the first orthogonality relation, we prove that the exponential generating function for the number of homomorphisms from a finitely generated group A to G_n is described as a sum of exponential functions of the form exp(f(x)) which are determined by elements cφ_m(A) of the factor group A/φ_m(A) of A by the intersection φ_m(A) of all kernels of homomorphisms from A to C_m.
d)We obtain, p-adic properties of the exponential generating function of the number of homomorphisms from a finite cyclic p-group to the wreath product of a finite cyclic group of order p by the symmetric group of n-letters.
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Research Products
(10 results)
