Research on generating functions of the number of group homomorphisms and subgroup lattices of groups

得到了以下结果：1。设A是一个有限循环p群，设G是A作用于的有限p群。如果半直积AG中G的补数不能被p|A|的最大公约数gcd（p|A|, |G|）整除，其中|A|为A阶，|G|为G的阶，则G是一个例外p群，即循环p群、二面体、广义四元数或半二面体2群。（该结果是与村井雅文共同研究得出的。）设A为两个环p群的直积，设G为A作用于其上的有限例外p群。G在AG中的补体数可以被|A|和|G|的gcd（|A|, |G|）整除。（该结果是与浅井恒信和Niwasaki Takashi共同研究得出的。）设A是循环p群与p^2阶循环群的直积，设G是A作用于的有限p群。G在AG中的补数可被gcd（|A|, |G|）整除，这是对P. hall的一个定理的推广。对于有限阿贝尔p群的若干个置换表示，得到了它的p进性质。在证明中，应用了生成函数。

The following results have been obtained.1. Let A be a finite cyclic p-group, and let G be a finite p-group on which A acts. If the number of complements of G in the semi-direct product AG is not divisible by the greatest common divisor gcd(p|A|, |G|) of p|A|, where |A| is the order of A, and the order |G| of G, then G is a exceptional p-group, namely, a cyclic p-group, a dihedral, generalized quaternion, or semi-dihedral 2-group. (This result has been obtained in the joint work with Masafumi Murai.)2. Let A be the direct product of two cyclic p-groups, and let G be a finite exceptional p-group on which A acts. The number of complements of G in AG is divisible by gcd(|A|, |G|) of |A| and |G|. (This result has been obtained in the joint work with Tsunenobu Asai and Takashi Niwasaki.)3. Let A be the direct product of a cyclic p-group and a cyclic group of order p^2, and let G be a finite p-group on which A acts. The number of complements of G in AG is divisible by gcd(|A|, |G|), which is a generalization of a theorem of P. Hall.4. For the number of permutation representations of a finite abelian p-group, its p-adic properties have been obtained. In the proof, the generating function has been applied.