Research on properties of generating functions of the number of special permutation representations and their applications.

a)研究了方程x^d=1的n个字母交替群解的指数生成函数中出现的两个p进幂级数exp（f(x)）和exp（g(x)）的性质。此外，当exp（f(x)）和exp（g(x)）分别作为序列{h_n}和{r_n}的指数生成函数时，我们用p进解析函数表示h_n和r_n。b)设H为有限单群，并将H与H的所有内自同构组成的群等价。设G为H的一组自同构，设G的元素a不是H的元素，且a^2是H的元素。"如果一个H阶的除数e是2除以H阶的最大幂的倍数，如果方程x^<2e>=1在aH中的解的个数是e，c)我们认为G_n是有限群G与n个字母的对称群的环积同态的核，该群是由复数中单位的一个本元的m-根生成的G到循环群C_m的同态决定的。利用第一正交关系，证明了有限生成群a到G_n的同态数的指数生成函数被描述为exp（f(x)）形式的指数函数的和，这些指数函数由因子群a / a的φ_m(a)的元素cφ_m(a)由a到C_m的所有同态核的φ_m(a)的交决定。d)得到了有限循环p群到p阶有限循环群环积的同态数的指数生成函数的p进性质。

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.