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2002 Fiscal Year Final Research Report Summary

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

Research Project

Project/Area Number 13640004
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionMURORAN INSTITUTE OF TECHNOLOGY

Principal Investigator

TAKEGAHARA Yugen  Muroran institute of Techinology; Faculty of Engineering, Professor, 工学部, 教授 (10211351)

Co-Investigator(Kenkyū-buntansha) CHIGIRA Naoki  Muroran institute of Techinology; Faculty of Engineering, Associate Professor, 工学部, 助教授 (40292073)
SATO Motohiko  Muroran institute of Techinology; Faculty of Engineering, Associate Professor, 工学部, 助教授 (30254139)
Project Period (FY) 2001 – 2002
Keywordsfinite p-group / semi-direct product / complement / exceptional p-group / abelian group / permutation representation / p-adic property / generating function
Research Abstract

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.

  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] T.Asai, Y.Takegahara: "|Hom(A, G)|, IV"Journal of Algebra. 246. 543-563 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Takegahara: "On P. Hall's relations in finite groups"RIMS Kokyuroku. 1214. 27-36 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Asai, Y.Takegahara, T.Niwasaki: "On the number of cocycles for dihedral groups"RIMS Kokyuroku. 1251. 70-82 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Motohiko Sato: "A level set approach to semi-continuous viscosity solutions for Cauchy problems"Communications in P.D.E.. 26. 813-839 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Asai, T.Niwasaki, Y.Takegahara: "Crossed homomorphisms from rank 2 abelian to exceptional p-groups"Journal of Algebra. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Asai, Y. Takegahara: "|Hom (A, G) |, IV"Journal of Algebra. 246. 543-563 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] 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)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Takegahara: "On P. Hall's relations in finite groups"RIMS Kokyuroku. 1214. 27-36 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Takegahara: "A generating function for the number of homomorphisms from a finitely generated abelian group to an alternating group"Journal Algebra. 248. 554-574 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Asai, Y. Takegahara, T. Niwasaki: "On the number of cocycles for dihedral groups"RIMS Kokyuroku. 1251. 70-82 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Motohiko Sato: "A level set approach to semi-continuous viscosity solutions for Cauchy problems"Communications in P. D. E.. 26. 813-839 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Asai, T. Niwasaki, Y. Takegahara: "Crossed homomorphisms from rank 2 abelian to exceptional p-groups"Journal of Algebra. to appear.

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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