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

a)对于每个自然数n，设G_n为有限群与对称群S_n在n个字母上的圈积的某些线性特征的核。当A是有限生成群或无限生成群时，确定A到G_n的同态个数的指数生成函数。有限群G b),函数的性质从G T_n非负整数,对于每一个G, T_n (G)是序列的数量(x_1、x_2…,x_n) G的元素满足更高的换向器(x_1、x_2…,x_n) = G obtained.c)是获得一个有限群G的幂零类n当且仅当一个特定的字符表的矩阵确定G是幂零指数无日期)一定的同余方程模p子群的指数d的数量,得到了有限阿贝尔群自由积中的一个固定自然数D。e)设有限群a是有限群G的算子群，并设G为右a集。一个自由的右G集Y称为（A，G）-集，如果Y是一个左A集，其作用为A (yg)=(ay)^ag, A∈A, Y∈Y, G∈G。如果一个（A，G）集是可传递的A集，则称其为简单集。确定了简单（A，G）-集的同构类的代表的完备集。定义了（A，G）-集合范畴的Grothendieck环，推广了有限群Burnside环的一些性质。此外，作为该理论的应用，得到了Brauer归纳定理的一个显式公式。

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.