p-Blocks Relative to a Character of a Normal Subgroup II


Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let \(\theta \) be a G-invariant irreducible character of N. In Rizo (J Algebra 514:254–272, 2018), we introduced a canonical partition of the set \(\mathrm{Irr}(G|\theta )\) of irreducible constituents of the induced character \(\theta ^G\), relative to the prime p. We call the elements of this partition the \(\theta \)-blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on \(\{ x \in G \, |\, x_p \in N\}\), which supersedes previous non-canonical constructions. This allows us to define \(\theta \)-decomposition numbers in a natural way. We also prove that the elements of the partition of \({\text {Irr}}(G|\theta )\) established by these \(\theta \)-decomposition numbers are the \(\theta \)-blocks.

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  1. 1.

    Conlon, S.B.: Twisted group algebras and their representations. J. Austral. Math. Soc. 4, 152–173 (1964)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Isaacs, I.M.: Characters of solvable and symplectic groups. Am. J. Math. 95, 594–635 (1973)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Isaacs, I.M.: Character Theory of Finite Groups. Pure and Applied Mathematics, No. 69. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1976)

    Google Scholar 

  4. 4.

    Külshammer, B., Robinson, G.R.: Characters of relatively projective modules II. J. Lond. Math. Soc. 36, 59–67 (1987)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Navarro, G.: Characters and Blocks of Finite Groups. London Mathematical Society Lecture Note Series, vol. 250. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  6. 6.

    Navarro, G.: Brauer characters relative to a normal subgroup. Proc. Lond. Math. Soc. (3) 81(1), 55–71 (2000)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Navarro, G.: Bases for class functions on finite groups, Biblioteca de la Revista Matemática Iberoamericana, In: Proceedings Encuentro en Teoria de Grupos y sus Aplicaciones (Zaragoza, 2011), pp. 225–232 (2012)

  8. 8.

    Navarro, G.: Character Theory and the McKay Conjecture. Cambridge Studies in Advanced Mathematics 175. Cambridge (2018)

  9. 9.

    Reynolds, W.F.: Block idempotents of twisted group algebras. Proc. Am. Math. Soc. 17, 280–282 (1966)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Rizo, N.: \(p\)-Blocks relative to a character of a normal subgroup. J. Algebra 514, 254–272 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Zeng, J.: On relatively projective modules and the Külshammer–Robinson basis. Commun. Algebra 31, 2425–2443 (2003)

    Article  Google Scholar 

Download references


Most of this paper is part of my Ph.D. thesis under the direction of Gabriel Navarro. I would like to take this opportunity to warmly thank him for thoroughly reading this manuscript. I would also like to thank Radha Kessar and Charles Eaton for many useful comments on this work and, in general, on my thesis.

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Correspondence to Noelia Rizo.

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The author acknowledges support by Ministerio de Ciencia e Innovación PID2019-103854GB-I00 and FEDER funds.

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Rizo, N. p-Blocks Relative to a Character of a Normal Subgroup II. Mediterr. J. Math. 17, 171 (2020). https://doi.org/10.1007/s00009-020-01599-z

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  • Brauer p-blocks
  • Brauer’s first main theorem
  • Block orthogonality
  • Central extensions

Mathematics Subject Classification

  • Primary 20D
  • Secondary 20C15