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Metacyclic Defect Groups

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2127)

Abstract

As the first application of the general methods we classify blocks with metacyclic defect groups. Here in case p = 2 we obtain an almost complete result due to various authors. In the odd case we give a proof of Brauer’s k(B)-Conjecture, Olsson’s Conjecture and Brauer’s Height Zero Conjecture. Moreover, we use a recent result by Watanabe to describe blocks with metacyclic, minimal non-abelian defect groups.

Keywords

  • Defect Group
  • Finite Simple Group
  • Principal Block
  • Quaternion Defect
  • Quasisimple Group

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. An, J.: Controlled blocks of the finite quasisimple groups for odd primes. Adv. Math. 227(3), 1165–1194 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Brauer, R.: Investigations on group characters. Ann. Math. (2) 42, 936–958 (1941)

    Google Scholar 

  3. Brauer, R.: Some applications of the theory of blocks of characters of finite groups. II. J. Algebra 1, 307–334 (1964)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Brauer, R.: On 2-blocks with dihedral defect groups. In: Symposia Mathematica, vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), pp. 367–393. Academic, London (1974)

    Google Scholar 

  5. Cabanes, M., Picaronny, C.: Types of blocks with dihedral or quaternion defect groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39(1), 141–161 (1992). Revised version: http://www.math.jussieu.fr/~cabanes/type99.pdf

  6. Craven, D.A., Eaton, C.W., Kessar, R., Linckelmann, M.: The structure of blocks with a Klein four defect group. Math. Z. 268(1–2), 441–476 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Craven, D.A., Glesser, A.: Fusion systems on small p-groups. Trans. Am. Math. Soc. 364(11), 5945–5967 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Dade, E.C.: Blocks with cyclic defect groups. Ann. Math. (2) 84, 20–48 (1966)

    Google Scholar 

  9. Dietz, J.: Stable splittings of classifying spaces of metacyclic p-groups, p odd. J. Pure Appl. Algebra 90(2), 115–136 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Donovan, P.W.: Dihedral defect groups. J. Algebra 56(1), 184–206 (1979)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Eaton, C.W., Kessar, R., Külshammer, B., Sambale, B.: 2-Blocks with abelian defect groups. Adv. Math. 254, 706–735 (2014)

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. Erdmann, K.: Blocks whose defect groups are Klein four groups: a correction. J. Algebra 76(2), 505–518 (1982)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Erdmann, K.: Blocks of Tame Representation Type and Related Algebras. Lecture Notes in Mathematics, vol. 1428. Springer, Berlin (1990)

    Google Scholar 

  14. Feit, W.: The Representation Theory of Finite Groups. North-Holland Mathematical Library, vol. 25. North-Holland Publishing, Amsterdam (1982)

    Google Scholar 

  15. Gao, S.: On Brauer’s k(B)-problem for blocks with metacyclic defect groups of odd order. Arch. Math. (Basel) 96(6), 507–512 (2011)

    Google Scholar 

  16. Gao, S.: Blocks of full defect with nonabelian metacyclic defect groups. Arch. Math. (Basel) 98, 1–12 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Gluck, D.: Rational defect groups and 2-rational characters. J. Group Theory 14(3), 401–412 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. Hendren, S.: Extra special defect groups of order p 3 and exponent p 2. J. Algebra 291(2), 457–491 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. Héthelyi, L., Külshammer, B.: Characters, conjugacy classes and centrally large subgroups of p-groups of small rank. J. Algebra 340, 199–210 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Holloway, M., Koshitani, S., Kunugi, N.: Blocks with nonabelian defect groups which have cyclic subgroups of index p. Arch. Math. (Basel) 94(2), 101–116 (2010)

    Google Scholar 

  21. Holm, T.: Blocks of tame representation type and related algebras: derived equivalences and hochschild cohomology. Habilitationsschrift, Magdeburg (2001)

    Google Scholar 

  22. Holm, T.: Notes on donovan’s conjecture for blocks of tame representation type (2014). http://www.iazd.uni-hannover.de/~tholm/ARTIKEL/donovan.ps

  23. Holm, T., Kessar, R., Linckelmann, M.: Blocks with a quaternion defect group over a 2-adic ring: the case \(\tilde{A}_{4}\). Glasg. Math. J. 49(1), 29–43 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. Horimoto, H., Watanabe, A.: On a perfect isometry between principal p-blocks of finite groups with cyclic p-hyperfocal subgroups (2012). Preprint

    Google Scholar 

  25. Isaacs, I.M., Navarro, G.: New refinements of the McKay conjecture for arbitrary finite groups. Ann. Math. (2) 156(1), 333–344 (2002)

    Google Scholar 

  26. Kessar, R.: Introduction to block theory. In: Group Representation Theory, pp. 47–77. EPFL Press, Lausanne (2007)

    Google Scholar 

  27. Kessar, R., Linckelmann, M.: On perfect isometries for tame blocks. Bull. Lond. Math. Soc. 34(1), 46–54 (2002)

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. Kessar, R., Linckelmann, M., Navarro, G.: A characterisation of nilpotent blocks (2014). arXiv:1402.5871v1

  29. Kiyota, M.: On 3-blocks with an elementary abelian defect group of order 9. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31(1), 33–58 (1984)

    MATH  MathSciNet  Google Scholar 

  30. Koshitani, S.: Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups. Commun. Algebra 31(5), 2229–2243 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  31. Koshitani, S., Miyachi, H.: Donovan conjecture and Loewy length for principal 3-blocks of finite groups with elementary abelian Sylow 3-subgroup of order 9. Commun. Algebra 29(10), 4509–4522 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

  32. Liedahl, S.: Enumeration of metacyclic p-groups. J. Algebra 186(2), 436–446 (1996)

    CrossRef  MATH  MathSciNet  Google Scholar 

  33. Linckelmann, M.: Derived equivalence for cyclic blocks over a P-adic ring. Math. Z. 207(2), 293–304 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

  34. Linckelmann, M.: A derived equivalence for blocks with dihedral defect groups. J. Algebra 164(1), 244–255 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  35. Linckelmann, M.: The source algebras of blocks with a Klein four defect group. J. Algebra 167(3), 821–854 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  36. Linckelmann, M.: Fusion category algebras. J. Algebra 277(1), 222–235 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

  37. Olsson, J.B.: On 2-blocks with quaternion and quasidihedral defect groups. J. Algebra 36(2), 212–241 (1975)

    CrossRef  MATH  MathSciNet  Google Scholar 

  38. Park, S.: The gluing problem for some block fusion systems. J. Algebra 323(6), 1690–1697 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  39. Puig, L., Usami, Y.: Perfect isometries for blocks with abelian defect groups and Klein four inertial quotients. J. Algebra 160(1), 192–225 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  40. Puig, L., Usami, Y.: Perfect isometries for blocks with abelian defect groups and cyclic inertial quotients of order 4. J. Algebra 172(1), 205–213 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  41. Rickard, J.: Derived categories and stable equivalence. J. Pure Appl. Algebra 61(3), 303–317 (1989)

    CrossRef  MATH  MathSciNet  Google Scholar 

  42. Robinson, G.R.: Weight conjectures for ordinary characters. J. Algebra 276(2), 761–775 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

  43. Robinson, G.R.: Large character heights, Qd(p), and the ordinary weight conjecture. J. Algebra 319(2), 657–679 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  44. Robinson, G.R.: On the focal defect group of a block, characters of height zero, and lower defect group multiplicities. J. Algebra 320(6), 2624–2628 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  45. Rouquier, R.: The derived category of blocks with cyclic defect groups. In: Derived Equivalences for Group Rings. Lecture Notes in Mathematics, vol. 1685, pp. 199–220. Springer, Berlin (1998)

    Google Scholar 

  46. Sambale, B.: 2-blöcke mit metazyklischen und minimal nichtabelschen defektgruppen. Dissertation, Südwestdeutscher Verlag für Hochschulschriften, Saarbrücken (2011)

    Google Scholar 

  47. Sambale, B.: Fusion systems on metacyclic 2-groups. Osaka J. Math. 49, 325–329 (2012)

    MATH  MathSciNet  Google Scholar 

  48. Sambale, B.: Brauer’s Height Zero Conjecture for metacyclic defect groups. Pac. J. Math. 262(2), 481–507 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  49. Sambale, B.: The Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups. Proc. Am. Math. Soc. (2014, to appear). arXiv:1403.5153v1

  50. Schulz, N.: Über p-blöcke endlicher p-auflösbarer Gruppen. Dissertation, Universität Dortmund (1980)

    Google Scholar 

  51. Stancu, R.: Control of fusion in fusion systems. J. Algebra Appl. 5(6), 817–837 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  52. Uno, K.: Dade’s conjecture for tame blocks. Osaka J. Math. 31(4), 747–772 (1994)

    MATH  MathSciNet  Google Scholar 

  53. Usami, Y.: On p-blocks with abelian defect groups and inertial index 2 or 3. I. J. Algebra 119(1), 123–146 (1988)

    CrossRef  MATH  MathSciNet  Google Scholar 

  54. Webb, P.: An introduction to the representations and cohomology of categories. In: Group Representation Theory, pp. 149–173. EPFL Press, Lausanne (2007)

    Google Scholar 

  55. Yang, S.: On Olsson’s conjecture for blocks with metacyclic defect groups of odd order. Arch. Math. (Basel) 96, 401–408 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  56. Yang, S.: 3-Blocks with Abelian defect groups isomorphic to \(Z_{3^{m}} \times Z_{3^{n}}\). Acta Math. Sin. (Engl. Ser.) 29(12), 2245–2250 (2013)

    Google Scholar 

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Sambale, B. (2014). Metacyclic Defect Groups. In: Blocks of Finite Groups and Their Invariants. Lecture Notes in Mathematics, vol 2127. Springer, Cham. https://doi.org/10.1007/978-3-319-12006-5_8

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