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Parametrization of central Frattini extensions and isomorphisms of small group rings

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Abstract

We solve the modular isomorphism problem for small group rings, i.e., we determine, for a given finite p-group H, precisely which central Frattini extensions of H give rise to isomorphic small group rings over the field with p elements.

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References

  1. J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics, Vol. 11, Cambridge University Press, Cambridge, 1986, Modular representations as an introduction to the local representation theory of finite groups.

    Google Scholar 

  2. I. C. Borge and O. A. Laudal, The modular isomorphism problem, University of Oslo, Department of Mathematics, Preprint series in Pure Mathematics 2002, no. 19, ISBN 82-553-1357-5, (http://www.math.uio.no/eprint/).

  3. I. C. Borge and O. A. Laudal, The modular isomorphism problem, University of Oslo, Department of Mathematics, Preprint series in Pure Mathematics 2003, no. 25, ISBN 82-553-1393-1, (http://www.math.uio.no/eprint/).

  4. J. F. Carlson, Periodic modules over modular group algebras, Journal of the London Mathematical Society (2) 15 (1977), 431–436.

    Article  MATH  Google Scholar 

  5. D. B. Coleman, On the modular group ring of a p-group, Proceedings of the American Mathematical Society 15 (1964), 511–514.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. W. Curtis and I. Reiner, Methods of Representation Theory. Vol. I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1990, With applications to finite groups and orders, Reprint of the 1981 original, A Wiley-Interscience Publication.

    Google Scholar 

  7. The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.3, package SmallGroups, 2002, (http://www.gap-system.org).

  8. M. Hertweck and M. Soriano, Central p-group extensions: Laudal obstruction spaces revisited, submitted, 2005.

  9. E. T. Hill, The annihilator of radical powers in the modular group ring of a p-group, Proceedings of the American Mathematical Society 25 (1970), 811–815.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York, 1971.

    MATH  Google Scholar 

  11. B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin, 1967.

    MATH  Google Scholar 

  12. G. D. James, Representations of general linear groups, London Mathematical Society Lecture Note Series, Vol. 94, Cambridge University Press, Cambridge, 1984.

    MATH  Google Scholar 

  13. S. A. Jennings, The structure of the group ring of a p-group over a modular field, Transactions of the American Mathematical Society 50 (1941), 175–185.

    Article  MATH  MathSciNet  Google Scholar 

  14. I. B. S. Passi and S. K. Sehgal, Isomorphism of modular group algebras, Mathematische Zeitschrift 129 (1972), 65–73.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. S. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley-Interscience, New York, 1977.

    Google Scholar 

  16. J. Ritter and S. Sehgal, Isomorphism of group rings, Archiv der Mathematik (Basel) 40 (1983), 32–39.

    MATH  MathSciNet  Google Scholar 

  17. D. J. S. Robinson, A Course in the Theory of Groups, second ed., Graduate Texts in Mathematics, Vol. 80, Springer-Verlag, New York, 1996.

    Google Scholar 

  18. K. Roggenkamp and L. Scott, Isomorphisms of p-adic group rings, Annals of Mathematics (2) 126 (1987), 593–647.

    Article  MathSciNet  Google Scholar 

  19. F. Röhl, On the isomorphism problem for group rings and completed augmentation ideals, Rocky Mountain Journal of Mathematics 17 (1987), 853–863.

    Article  MATH  MathSciNet  Google Scholar 

  20. F. Röhl, On automorphisms of complete algebras and the isomorphism problem for modular group rings, Canadian Journal of Mathematics 42 (1990), 383–394.

    MATH  Google Scholar 

  21. R. Sandling, The isomorphism problem for group rings: a survey, in Orders and their Applications (Oberwolfach, 1984), Lecture Notes in Mathematics, Vol. 1142, Springer, Berlin, 1985, pp. 256–288.

    Chapter  Google Scholar 

  22. H. N. Ward, Some results on the group algebra of a group over a prime field, Seminar in Group Theory, Harvard University, Cambridge, MA, 1960–61, Mimeographed Notes, pp. 13–19.

    Google Scholar 

  23. A. Whitcomb, The group ring problem, Ph.D. thesis, University of Chicago, 1968.

  24. H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe einen Lie-Ring mit der Charakteristik p zuzuordnen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 13 (1939), 200–207.

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Institut für Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, 70550, Stuttgart, Germany

    Martin Hertweck

  2. Institut für Mathematik, Lehrgebiet Algebra und Zahlentheorie, Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Marcos Soriano

Authors
  1. Martin Hertweck
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  2. Marcos Soriano
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The first author acknowledges support by the Deutsche Forschungsgemeinschaft.

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Hertweck, M., Soriano, M. Parametrization of central Frattini extensions and isomorphisms of small group rings. Isr. J. Math. 157, 63–102 (2007). https://doi.org/10.1007/s11856-006-0003-5

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  • DOI: https://doi.org/10.1007/s11856-006-0003-5

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