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

A)研究了关于方程x^d=1的n个字母的交错群的解的个数的指数母函数中的两个p元幂函数exp(f(X))和exp(g(X))的性质.此外，当exp(f(X))和exp(g(X))分别被认为是序列{h_n}和{r_n}的指数母函数时，我们用p元解析函数表示h_n和r_n.设G是H的一个自同构群，设G的一个元素a不是H的一个元素，a^2是H的一个元素，猜想是：“如果H阶除数e是除以H阶的2的最大幂的倍数，且若方程x^&lt的解的个数；2e&gt；=1是e，则方程x^&lt；2e&gt；=1“的每个元素都是方程x^&lt；2e&gt；=1”的解.c)我们认为G_n是有限群G与n字母对称群的圈积的同态的核，该同态由G到由复数的单位本原m次根生成的循环群Cm的同态决定.利用第一正交关系，我们证明了有限生成群A到G_n的同态个数的指数母函数是由A的因子群A的元素cφ_m(A)/A的φ_m(A)由从A到C_m的同态的所有核的交集φ_m(A)决定的指数函数之和。有限循环p-群到p阶有限循环群的圈积与n字母对称群的同态的个数的指数母函数的p-进的性质。

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.