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A Remark on the Construction of Centric Linking Systems

  • B. Oliver
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 242)

Abstract

We give examples to show that it is not, in general, possible to prove the existence and uniqueness of centric linking systems associated to a given fusion system inductively by adding one conjugacy class at a time to the categories. This helps to explain why it was so difficult to prove that these categories always exist, and also helps to motivate the procedure used by Chermak [5] when he did prove it. It also shows that the claim by Puig [12] to have proven this result in a recently published paper is not correct.

Keywords

Finite groups Sylow subgroups Fusion 

2000 Mathematics Subject Classification

Primary 20D20 Secondary 20D45 20E45 

References

  1. 1.
    M. Aschbacher, R. Kessar, & B. Oliver, Fusion systems in algebra and topology, Cambridge Univ. Press (2011).CrossRefGoogle Scholar
  2. 2.
    C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of \(p\)-completed classifying spaces of finite groups, Invent. Math., 151 (2003), 611–664.MathSciNetCrossRefGoogle Scholar
  3. 3.
    C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, Journal Amer. Math. Soc. 16 (2003), 779–856.MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley (1985).Google Scholar
  5. 5.
    A. Chermak, Fusion systems and localities, Acta Math. 211 (2013), 47–139.MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Glauberman & J. Lynd, Control of weak closure and existence and uniqueness of centric linking systems, Invent. Math. 206 (2016), 441–484.MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Grodal, Higher limits via subgroup complexes, Annals of Math., 155 (2002), 405–457.MathSciNetCrossRefGoogle Scholar
  8. 8.
    S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-maps of \(BG\) via \(G\)-actions, Annals of Math., 135 (1992), 183–270.MathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Oliver, Equivalences of classifying spaces completed at the prime two, Memoirs Amer. Math. Soc. 848 (2006).Google Scholar
  10. 10.
    B. Oliver, Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory, Acta Math. 211 (2013), 141–175.MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Puig, Frobenius categories, J. Algebra 303 (2006), 309–357.MathSciNetCrossRefGoogle Scholar
  12. 12.
    L. Puig, Existence, uniqueness, and functoriality of the perfect locality over a Frobenius \(P\)-category, Algebra Coll. 23 (2016), 541–622.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.LAGAUniversité Paris 13, Sorbonne Paris Cité, UMR 7539 du CNRSVilletaneuseFrance

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