Advertisement

When the Group Ring of a Finite Simple Group is Serial

  • A. KukharevEmail author
  • I. Kaygorodov
  • I. Gorshkov
Article
  • 2 Downloads

A ring is said to be serial if its right and left regular modules are the direct sums of chain modules. The aim of the paper is to give an answer to the following question: for which finite simple groups, the group ring over a given field is serial.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. L. Alperin, Local Representation Theory, Cambridge Univ. Press (1989).Google Scholar
  2. 2.
    Y. Baba and K. Oshiro, Classical Artinian Rings and Related Topics, World Scientific Publ. (2009).Google Scholar
  3. 3.
    Y. Benson, Representations and Cohomology. I, Cambridge Stud. Adv. Math., 30 (1995).Google Scholar
  4. 4.
    H. I. Blau, “On Brauer stars,” J. Algebra, 90, 169–188 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    W. Bosma, J. Cannon, and C. Playoust, “The MAGMA algebra system I: The user language,” J. Symbolic Comput., 24, 235–265 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. Burkhardt, “Die Zerlegungsmatrizen der Gruppen PSL(2, p f ),” J. Algebra, 40, 75–96 (1976).Google Scholar
  7. 7.
    R. Burkhardt, “Über die Zerlegungszahlen der Suzukigruppen Sz(q),” J. Algebra, 59, No. 2, 421–433 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. H. Conway (et al.), Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press (1985).Google Scholar
  9. 9.
    H. Dietrich, C. R. Leedham-Green, F. Lübeck, and E. A. O’Brien, “Constructive recognition of classical groups in even characteristic,” J. Algebra, 391, 227–255 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Eisenbud and P. Griffith, “Serial rings,” J. Algebra, 17, 389–400 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    W. Feit, The Representation Theory of Finite Groups, North-Holland Math. Library, 25 (1982).Google Scholar
  12. 12.
    W. Feit, “Possible Brauer trees,” Illinois J. Math., 28, 43–56 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. Fong and B. Srinivasan, “Brauer trees in classical groups,” J. Algebra, 131, 179–225 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.6, 2016, http://www.gap-system.org.
  15. 15.
    M. Geck, “Irreducible Brauer characters of the 3-dimensional unitary group in non-defining characteristic,” Algebra, 18, No. 2, 563–584 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    R. Gow, “Products of two involutions in classical groups of characteristic 2,” J. Algebra, 71, 583–591 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    D. G. Higman, “Indecomposable representations at characteristic p,” Duke Math. J., 21, 377–381 (1954).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    G. Hiss, “The Brauer trees of the Ree groups,” Algebra, 19(3), 871–888 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    G. Hiss and K. Lux, Brauer Trees of Sporadic Groups, Clarendon Press, Oxford (1989).zbMATHGoogle Scholar
  20. 20.
    G. J. Janusz, “Indecomposable modules for finite groups,” Annals Math., 89, 209–241 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Kukharev and G. Puninski, “Serial group rings of finite groups. p-nilpotency,” Zap. Nauchn. Semin. POMI, 413, 134–152 (2013).zbMATHGoogle Scholar
  22. 22.
    A. Kukharev and G. Puninski, “Serial group rings of finite groups. p-solvability,” Algebra Discr. Math., 16, 201–216 (2013).MathSciNetzbMATHGoogle Scholar
  23. 23.
    A. Kukharev and G. Puninski, “The seriality of group rings of alternating and symmetric groups,” Vestnik BGU, Math.-Inform. Ser., 2, 61–64 (2014).Google Scholar
  24. 24.
    A. Kukharev and G. Puninski, “Serial group rings of finite groups. Simple sporadic groups and Suzuki groups,” Zap. Nauchn. Semin. POMI, 435, 73–94 (2015).Google Scholar
  25. 25.
    A. Kukharev and G. Puninski, “Serial group rings of finite groups. General linear and close groups,” Algebra Discrete Math., 20, No. 1, 259–269 (2015).MathSciNetzbMATHGoogle Scholar
  26. 26.
    A. Kukharev and G. Puninski, “Serial group rings of classical groups defined over fields with odd number of elements,” Zap. Nauchn. Semin. POMI, 452, 158–176 (2016).Google Scholar
  27. 27.
    A. Kukharev and G. Puninski, “Serial group rings of finite groups of Lie type,” Fundam. Appl. Math., 21, 135–144 (2016).zbMATHGoogle Scholar
  28. 28.
    A. V. Kukharev, “Seriality of group rings of unimodular projective groups,” in: Proc. 71st Student Conf. of Belarusian State Univ., Vol. 1, Minsk (2014), pp. 11–14.Google Scholar
  29. 29.
    K. Lux and H. Pahlings, Representations of Groups. A Computational Approach, Cambridge Studies in Advanced Mathematics, 124 (2010).Google Scholar
  30. 30.
    K. Morita, “On group rings over a modular field which possess radicals expressible as principal ideals,” Sci. Repts. Tokyo Daigaku, 4, 177–194 (1951).MathSciNetzbMATHGoogle Scholar
  31. 31.
    N. Naerig, “A construction of almost all Brauer trees,” J. Group Theory, 11, 813–829 (2008).Google Scholar
  32. 32.
    G. Puninski, Serial Rings, Kluwer (2001).Google Scholar
  33. 33.
    G. R. Robinson, “Some uses of class algebra constants,” J. Algebra, 91, 64–74 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    M. Sawabe, “A note on finite simple groups with Abelian Sylow p-subgroups,” Tokyo Math. J., 30, 293–304 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    M. Sawabe and A. Watanabe, “On the principal blocks of finite groups with Abelian Sylow p-subgroups,” J. Algebra, 237, 719–734 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    J. Scopes, “Cartan matrices and Morita equivalence for blocks of the symmetric groups,” J. Algebra, 142, 441–455 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    B. Srinivasan, “On the indecomposable representations of a certain class of groups,” Proc. Lond. Math. Soc., 10, 497–513 (1960).MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    M. Stather, “Constructive Sylow theorems for the classical groups,” J. Algebra, 316, 536–559 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    A. A. Tuganbaev, Ring Theory, Arithmetical Rings and Modules, Independent University, Moscow (2009).Google Scholar
  40. 40.
    Yu. Volkov, A. Kukharev, and G. Puninski, “The seriality of the group ring of a finite group depends only of characteristic of the field,” Zap. Nauchn. Semin. POMI, 423, 57–66 (2014).zbMATHGoogle Scholar
  41. 41.
    H. Wielandt, “Sylowgruppen and Kompositions-Struktur,” Abhand. Math. Sem., Hamburg, 22, 215–228 (1958).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Vitebsk State University named after P.M. MasherovVitebskBelarus
  2. 2.Universidade Federal do ABCSanto AndréBrazil

Personalised recommendations