| Budget Amount *help |
¥3,730,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
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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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