Monatshefte für Mathematik

, Volume 166, Issue 3–4, pp 467–495 | Cite as

Transfers of metabelian p-groups

  • Daniel C. MayerEmail author


Explicit expressions for the transfers V i from a metabelian p-group G of coclass cc(G) = 1 to its maximal normal subgroups M 1, . . . , M p+1 are derived by means of relations for generators. The expressions for the exceptional case p = 2 differ significantly from the standard case of odd primes p ≥ 3. In both cases the transfer kernels Ker(V i ) are calculated and the principalisation type of the metabelian p-group is determined, if G is realised as the Galois group \({{\rm{Gal}}({F}_p^2(K)\vert K)}\) of the second Hilbert p-class field \({{F}_p^2(K)}\) of an algebraic number field K. For certain metabelian 3-groups G with abelianisation G/G′ of type (3, 3) and of coclass cc(G) = r ≥ 3, it is shown that the principalisation type determines the position of G on the coclass graph \({\mathcal{G}(3,r)}\) in the sense of Eick and Leedham-Green.


Metabelian p-groups of maximal class Transfers of 2-groups Tree of metabelian 3-groups of non-maximal class Principalisation of p-class groups Quadratic base fields 

Mathematics Subject Classification (2000)

Primary: 20F12 20F14 Secondary: 11R29 11R11 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artin E.: Beweis des allgemeinen Reziprozitätsgesetzes. Abh. Math. Sem. Univ. Hamburg 5, 353–363 (1927)zbMATHCrossRefGoogle Scholar
  2. 2.
    Artin E.: Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz. Abh. Math. Sem. Univ. Hamburg 7, 46–51 (1929)zbMATHCrossRefGoogle Scholar
  3. 3.
    Ascione, J.: On 3-groups of second maximal class. Ph.D. Thesis, Australian National University, Canberra (1979)Google Scholar
  4. 4.
    Benjamin E., Snyder C.: Real quadratic number fields with 2-class group of type (2,2). Math. Scand. 76, 161–178 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Berkovich Y.: Groups of prime power order, Volume 1 Expositions in Mathematics 46. de Gruyter, Berlin (2008)Google Scholar
  6. 6.
    Blackburn N.: On a special class of p-groups. Acta Math. 100, 45–92 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dietrich, H., Eick, B., Feichtenschlager, D.: Investigating p-groups by coclass with GAP. Computational group theory and the theory of groups, Contemp. Math., vol. 470, pp. 45–61. AMS, Providence (2008)Google Scholar
  8. 8.
    Eick B., Leedham-Green C.: On the classification of prime-power groups by coclass. Bull. Lond. Math. Soc. 40, 274–288 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gorenstein D.: Finite groups. Harper and Row, New York (1968)zbMATHGoogle Scholar
  10. 10.
    Hall P.: The classification of prime-power groups. J. Reine Angew. Math. 182, 130–141 (1940)MathSciNetGoogle Scholar
  11. 11.
    Hasse H.: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz. Jber. der DMV 6, 1–204 (1930)Google Scholar
  12. 12.
    Hilbert D.: Die Theorie der algebraischen Zahlkörper. Jber. der DMV 4, 175–546 (1897)Google Scholar
  13. 13.
    James R.: The groups of order p 6 (p an odd prime). Math. Comp. 34(150), 613–637 (1980)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kisilevsky H.: Some results related to Hilbert’s Theorem 94. J. Number Theory 2, 199–206 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kisilevsky H.: Number fields with class number congruent to 4 mod 8 and Hilbert’s Theorem 94. J. Number Theory 8, 271–279 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Leedham-Green C.R., McKay S.: The structure of groups of prime power order, London Math Soc Monographs, New Series, 27. Oxford Univ Press, Oxford (2002)Google Scholar
  17. 17.
    Mayer, D.C.: Principalization in complex S 3-fields. In: Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, Manitoba, Canada, 1990. Congressus Numerantium, vol. 80, pp. 73–87 (1991)Google Scholar
  18. 18.
    Mayer, D.C.: The second p-class group of a number field (in preparation)Google Scholar
  19. 19.
    Mayer, D.C.: Principalization algorithm via class group structure (in preparation)Google Scholar
  20. 20.
    Miech R.J.: Metabelian p-groups of maximal class. Trans. Am. Math. Soc. 152, 331–373 (1970)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Miyake K.: Algebraic investigations of Hilbert’s Theorem 94, the principal ideal theorem and the capitulation problem. Expo. Math. 7, 289–346 (1989)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Nebelung, B.: Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem. Inauguraldissertation, Band 1, Universität zu Köln (1989)Google Scholar
  23. 23.
    Nebelung, B.: Anhang zu Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem. Inauguraldissertation, Band 2, Universität zu Köln (1989)Google Scholar
  24. 24.
    Scholz A., Taussky O.: Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm. J. Reine Angew. Math. 171, 19–41 (1934)Google Scholar
  25. 25.
    Taussky O.: A remark concerning Hilbert’s Theorem 94. J. Reine Angew. Math. 239/240, 435–438 (1970)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.GrazAustria

Personalised recommendations