Research on properties of generating functions for permutation representations and their applications.

A)对每个自然数n，设G_n为对称群S_n关于n个字母的有限群的圈积的某些线性特征标的核。当A是有限生成群或准有限群时，确定了从A到G_n的同态个数的指数母函数。b)对于有限群G，T_n从G到非负整数的函数T_n的性质使得对于每个g，T_n(G)是序列(x_1，x_2，…)的个数满足高次交换子[x_1，x_2，…]的G的元素C)得到了有限群G是类n的幂零当且仅当由G的特征标表确定的某一矩阵为指标n的幂零.d)得到了有限交换群的自由积中指数d，d的子群个数的模p的同余方程.e)设有限群A是有限群G的算子群，则G是右A-集.自由右G-集合Y称为(A，G)-集合，如果Y是左A-集合，其作用由a(Yg)=(Ay)^ag，a∈A，y∈Y，g∈G给出，如果(A，G)-集合是传递A-集合，则称它为单集合。确定了单(A，G)-集的同构类的代表的完备集。定义了(A，G)-集范畴的Grothendieck环，推广了有限群的Burnside环的一些性质。此外，作为理论的应用，得到了Brauer归纳法定理的一个精确公式。

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.