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

群同态数生成函数及群子群格的研究

• 批准号: 13640004
• 负责人: TAKEGAHARA Yugen
• 金额: $ 1.92万
• 依托单位: MURORAN INSTITUTE OF TECHNOLOGY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640004/
• 关键词: finite p-group; semi-direct product; complement; exceptional p-group; abelian group; permutation representation; p-adic property; generating function; 対称群; 斜準同型; exceptional群; 準同型; 有限群; p-群; 環積

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.

取得了如下结果：1.设A是有限循环p-群，G是A作用于其上的有限p-群.如果半直积AG中G的补数不能被最大公约数gcd（p）整除|一|, |G|）P|一|得双曲余切值.|一|是A的顺序，|G|则G是例外p-群，即循环p-群、二面体、广义四元数或半二面体2-群。(This在与村井雅史的合作中取得了成果。2.设A是两个循环p-群的直积，G是A作用于其上的有限例外p-群. 2. G在AG中的补数可被gcd整除（|一|, |G|），共|一|和|G|. (This在与浅井恒信和Niwasaki隆的共同工作中获得了结果。3.设A是循环p-群与p ^2阶循环群的直积，G是A作用于其上的有限p-群。2. G在AG中的补数可被gcd整除（|一|, |G|），这是一个定理的推广P.Hall.4。对于有限交换p-群的置换表示数，得到了它的p-adic性质。在证明中，应用了母函数.

项目成果

Motohiko Sato: "A level set approach to semi-continuous viscosity solutions for Cauchy problems"Communications in P.D.E.. 26. 813-839 (2001)

Motohiko Sato：“柯西问题半连续粘度解决方案的水平集方法”Communications in P.D.E.. 26. 813-839 (2001)

Y.Takegahara: "A generating function for the number of homomorphisms from a finitely generated abelian group to an alternating group"Journal of Algebra. 248. 554-574 (2002)

Y.Takegahara：“从有限生成的阿贝尔群到交替群的同态数的生成函数”代数杂志。

T.Asai, T.Niwasaki, Y.Takegahara: "Crossed homomorphisms from rank 2 abelian to exceptional p-groups"Journal of Algebra. (to appear).

T.Asai、T.Niwasaki、Y.Takegahara：“从 2 阶阿贝尔到特殊 p 群的交叉同态”代数杂志。

H. Ishihara, H. Ochiai, Y. Takegahara, T. Yoshida: "p-divisibility of the number of solutions of x^p=1 in a symmetric group"Annals of Combinatorics. 5. 197-210 (2001)

H. Ishihara、H. Ochiai、Y. Takegahara、T. Yoshida：“对称群中 x^p=1 的解数的 p 整除性”组合学年鉴。

H.Ishihara: "p-divisibility of the number of solutions of x^p=1 in a symmetric group"Annals of Combinatorics. 5. 197-210 (2001)

H.Ishihara：“对称群中 x^p=1 的解数的 p 整除性”组合学年鉴。