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Reduction to Quasisimple Groups and the Classification

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Blocks of Finite Groups and Their Invariants

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

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Abstract

This chapter contains some reduction methods which often allows to answer a specific question about blocks by looking at blocks of quasisimple groups only. Most importantly, we introduce Fong’s reductions. In the second part we recall the classification of the finite simple groups and note some well-known results about representation theory of simple groups. Finally, we consider the p-solvable case.

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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)

    Article  MATH  MathSciNet  Google Scholar 

  2. An, J., Dietrich, H.: The AWC-goodness and essential rank of sporadic simple groups. J. Algebra 356, 325–354 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. An, J., Dietrich, H.: The essential rank of fusion systems of blocks of symmetric groups. Int. J. Algebra Comput. 22(1), 1250002, 15 (2012)

    Google Scholar 

  4. An, J., Dietrich, H.: The essential rank of Brauer categories for finite groups of lie type. Bull. Lond. Math. Soc. 45(2), 363–369 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. An, J., Eaton, C.W.: Modular representation theory of blocks with trivial intersection defect groups. Algebr. Represent. Theory 8(3), 427–448 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. An, J., Eaton, C.W.: Blocks with extraspecial defect groups of finite quasisimple groups. J. Algebra 328, 301–321 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. An, J., Eaton, C.W.: Nilpotent blocks of quasisimple groups for odd primes. J. Reine Angew. Math. 656, 131–177 (2011)

    MATH  MathSciNet  Google Scholar 

  8. An, J., Eaton, C.W.: Nilpotent blocks of quasisimple groups for the prime two. Algebr. Represent. Theory 16(1), 1–28 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Aschbacher, M., Kessar, R., Oliver, B.: Fusion Systems in Algebra and Topology. London Mathematical Society Lecture Note Series, vol. 391. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  10. Barker, L.: On p-soluble groups and the number of simple modules associated with a given Brauer pair. Q. J. Math. Oxford Ser. (2) 48(190), 133–160 (1997)

    Google Scholar 

  11. Bessenrodt, C., Olsson, J.B.: Spin representations and powers of 2. Algebr. Represent. Theory 3(3), 289–300 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Broué, M.: Isométries de caractères et équivalences de Morita ou dérivées. Inst. Hautes Études Sci. Publ. Math. 71, 45–63 (1990)

    Article  MATH  Google Scholar 

  13. Broué, M.: Equivalences of blocks of group algebras. In: Finite-Dimensional Algebras and Related Topics (Ottawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 424, pp. 1–26. Kluwer Academic Publishers, Dordrecht (1994)

    Google Scholar 

  14. Brunat, O., Gramain, J.B.: Perfect isometries and murnaghan-nakayama rules (2013). arXiv:1305.7449v1

  15. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: ATLAS of finite groups. Oxford University Press, Eynsham (1985). Maximal subgroups and ordinary characters for simple groups, With computational assistance from J.G. Thackray

    Google Scholar 

  16. Craven, D.A.: Control of fusion and solubility in fusion systems. J. Algebra 323(9), 2429–2448 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Craven, D.A.: The Theory of Fusion Systems. Cambridge Studies in Advanced Mathematics, vol. 131. Cambridge University Press, Cambridge (2011). An algebraic approach

    Google Scholar 

  18. Düvel, O.: On Donovan’s conjecture. J. Algebra 272(1), 1–26 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Eaton, C.W.: The equivalence of some conjectures of Dade and Robinson. J. Algebra 271(2), 638–651 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Eaton, C.W., Külshammer, B., Sambale, B.: 2-Blocks with minimal nonabelian defect groups ii. J. Group Theory 15, 311–321 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Eaton, C.W., Moretó, A.: Extending Brauer’s Height Zero Conjecture to blocks with nonabelian defect groups. Int. Math. Res. Not. 2014(20), 5581–5601 (2014)

    MATH  Google Scholar 

  22. Enguehard, M.: Isométries parfaites entre blocs de groupes symétriques. Astérisque 181–182, 157–171 (1990)

    MathSciNet  Google Scholar 

  23. Fong, P., Harris, M.E.: On perfect isometries and isotypies in finite groups. Invent. Math. 114(1), 139–191 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fong, P., Srinivasan, B.: The blocks of finite general linear and unitary groups. Invent. Math. 69(1), 109–153 (1982)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Gluck, D., Wolf, T.R.: Brauer’s height conjecture for p-solvable groups. Trans. Am. Math. Soc. 282(1), 137–152 (1984)

    MATH  MathSciNet  Google Scholar 

  27. Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups. Number 3. Part I. Chapter A. Mathematical Surveys and Monographs, vol. 40. American Mathematical Society, Providence (1998). Almost simple K-groups

    Google Scholar 

  28. Harris, M.E., Knörr, R.: Brauer correspondence for covering blocks of finite groups. Commun. Algebra 13(5), 1213–1218 (1985)

    Article  MATH  Google Scholar 

  29. Hiss, G.: Morita equivalences between blocks of finite chevalley groups. In: Proceedings of Representation Theory of Finite and Algebraic Groups, Osaka University, pp. 128–136 (2000)

    Google Scholar 

  30. Hiss, G., Kessar, R.: Scopes reduction and Morita equivalence classes of blocks in finite classical groups. J. Algebra 230(2), 378–423 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  31. Hiss, G., Kessar, R.: Scopes reduction and Morita equivalence classes of blocks in finite classical groups. II. J. Algebra 283(2), 522–563 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Hiss, G., Lux, K.: Brauer Trees of Sporadic Groups. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1989)

    MATH  Google Scholar 

  33. Humphreys, J.E.: Defect groups for finite groups of Lie type. Math. Z. 119, 149–152 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  34. Isaacs, I.M., Navarro, G.: Characters of p′-degree of p-solvable groups. J. Algebra 246(1), 394–413 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Jacobsen, M.W.: Block fusion systems of the alternating groups (2012). arXiv:1204.2702v1

  36. James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, vol. 16. Addison-Wesley Publishing, Reading (1981). With a foreword by P.M. Cohn, With an introduction by Gilbert de B. Robinson

    Google Scholar 

  37. Kessar, R.: Blocks and source algebras for the double covers of the symmetric and alternating groups. J. Algebra 186(3), 872–933 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kessar, R.: Scopes reduction for blocks of finite alternating groups. Q. J. Math. 53(4), 443–454 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kessar, R., Malle, G.: Quasi-isolated blocks and Brauer’s height zero conjecture. Ann. Math. (2) 178(1), 321–384 (2013)

    Google Scholar 

  40. Külshammer, B.: On p-blocks of p-solvable groups. Commun. Algebra 9(17), 1763–1785 (1981)

    Article  MATH  Google Scholar 

  41. Külshammer, B.: A remark on conjectures in modular representation theory. Arch. Math. (Basel) 49(5), 396–399 (1987)

    Google Scholar 

  42. Külshammer, B.: Donovan’s conjecture, crossed products and algebraic group actions. Israel J. Math. 92(1–3), 295–306 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  43. Külshammer, B., Puig, L.: Extensions of nilpotent blocks. Invent. Math. 102(1), 17–71 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  44. Landrock, P.: The non-principal 2-blocks of sporadic simple groups. Commun. Algebra 6(18), 1865–1891 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  45. Leedham-Green, C.R., Plesken, W.: Some remarks on Sylow subgroups of general linear groups. Math. Z. 191(4), 529–535 (1986)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  47. Moretó, A., Navarro, G.: Heights of characters in blocks of p-solvable groups. Bull. Lond. Math. Soc. 37(3), 373–380 (2005)

    Article  MATH  Google Scholar 

  48. Naehrig, M.: Die Brauer-Bäume des Monsters M in Charakteristik 29. Diplomarbeit, Aachen (2002)

    Google Scholar 

  49. Nagao, H.: On a conjecture of Brauer for p-solvable groups. J. Math. Osaka City Univ. 13, 35–38 (1962)

    MathSciNet  Google Scholar 

  50. Nagao, H., Tsushima, Y.: Representations of finite groups. Academic, Boston (1989). Translated from the Japanese

    MATH  Google Scholar 

  51. Navarro, G., Tiep, P.H.: Brauer’s height zero conjecture for the 2-blocks of maximal defect. J. Reine Angew. Math. 669, 225–247 (2012)

    MATH  MathSciNet  Google Scholar 

  52. Noeske, F.: Adgc for sporadic groups (2014). http://www.math.rwth-aachen.de/~Felix.Noeske/tabular.pdf

  53. Okuyama, T., Wajima, M.: Character correspondence and p-blocks of p-solvable groups. Osaka J. Math. 17(3), 801–806 (1980)

    MATH  MathSciNet  Google Scholar 

  54. Olsson, J.B.: McKay numbers and heights of characters. Math. Scand. 38(1), 25–42 (1976)

    MATH  MathSciNet  Google Scholar 

  55. Olsson, J.B.: Combinatorics and Representations of Finite Groups. Vorlesungen aus dem Fachbereich Mathematik der Universität GH Essen, Essen, vol. 20 (1993)

    Google Scholar 

  56. Puig, L.: Nilpotent extensions of blocks. Math. Z. 269(1–2), 115–136 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  57. Robinson, G.R.: Dade’s projective conjecture for p-solvable groups. J. Algebra 229(1), 234–248 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  58. Sawabe, M.: A note on finite simple groups with abelian Sylow p-subgroups. Tokyo J. Math. 30(2), 293–304 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  59. Sawabe, M., Watanabe, A.: On the principal blocks of finite groups with abelian Sylow p-subgroups. J. Algebra 237(2), 719–734 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  60. Schmid, P.: The Solution of the k(GV ) Problem. ICP Advanced Texts in Mathematics, vol. 4. Imperial College Press, London (2007)

    Google Scholar 

  61. Scopes, J.: Cartan matrices and Morita equivalence for blocks of the symmetric groups. J. Algebra 142(2), 441–455 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  62. The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.6.5 (2013). http://www.gap-system.org

  63. Usami, Y.: Perfect isometries for principal blocks with abelian defect groups and elementary abelian 2-inertial quotients. J. Algebra 196(2), 646–681 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  64. Waldmüller, R.: Untersuchungen zu donovans vermutung für klassische gruppen. Dissertation, Aachen (2005)

    Google Scholar 

  65. Weir, A.J.: Sylow p-subgroups of the classical groups over finite fields with characteristic prime to p. Proc. Am. Math. Soc. 6, 529–533 (1955)

    Google Scholar 

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  1. Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

    Benjamin Sambale

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Sambale, B. (2014). Reduction to Quasisimple Groups and the Classification. 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_7

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