Journal of the Mathematical Society of Japan

Cartan matrices and Brauer's $k(B)$-conjecture IV

Benjamin SAMBALE

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In this note we give applications of recent results coming mostly from the third paper of this series. It is shown that the number of irreducible characters in a $p$-block of a finite group with abelian defect group $D$ is bounded by $|D|$ (Brauer's $k(B)$-conjecture) provided $D$ has no large elementary abelian direct summands. Moreover, we verify Brauer's $k(B)$-conjecture for all blocks with minimal non-abelian defect groups. This extends previous results by various authors.

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J. Math. Soc. Japan, Volume 69, Number 2 (2017), 735-754.

First available in Project Euclid: 20 April 2017

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Primary: 20C15: Ordinary representations and characters
Secondary: 20C20: Modular representations and characters

blocks minimal non-abelian defect groups abelian defect groups Brauer's $k(B)$-conjecture


SAMBALE, Benjamin. Cartan matrices and Brauer's $k(B)$-conjecture IV. J. Math. Soc. Japan 69 (2017), no. 2, 735--754. doi:10.2969/jmsj/06920735.

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  • M. Aschbacher, R. Kessar and B. Oliver, Fusion systems in algebra and topology, London Math. Soc. Lecture Note Series, 391, Cambridge University Press, Cambridge, 2011.
  • M. Broué and L. Puig, A Frobenius theorem for blocks, Invent. Math., 56 (1980), 117–128.
  • C. W. Eaton, B. Külshammer and B. Sambale, $2$-Blocks with minimal nonabelian defect groups II, J. Group Theory, 15 (2012), 311–321.
  • W. Feit, The representation theory of finite groups, North-Holland Mathematical Library, 25, North-Holland Publishing Co., Amsterdam, 1982.
  • M. Fujii, On determinants of Cartan matrices of $p$-blocks, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 401–403.
  • S. Gao and J. Zeng, On the number of ordinary irreducible characters in a $p$-block with a minimal nonabelian defect group, Comm. Algebra, 39 (2011), 3278–3297.
  • The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.8; 2015, (
  • D. Gorenstein, Finite groups, Harper & Row Publishers, New York, 1968.
  • Z. Halasi and K. Podoski, Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc., 368 (2016), 5857–5887.
  • S. Hendren, Extra special defect groups of order $p^3$ and exponent $p^2$, J. Algebra, 291 (2005), 457–491.
  • S. Hendren, Extra special defect groups of order $p^3$ and exponent $p$, J. Algebra, 313 (2007), 724–760.
  • B. Huppert, Endliche Gruppen. I, Grundlehren der Mathematischen Wissenschaften, 134, Springer-Verlag, Berlin, 1967.
  • T. M. Keller and Y. Yang, Abelian quotients and orbit sizes of finite groups, arXiv:1407.6436v1.
  • R. Kessar and M. Linckelmann, On perfect isometries for tame blocks, Bull. London Math. Soc., 34 (2002), 46–54.
  • R. Kessar and G. Malle, Quasi-isolated blocks and Brauer's height zero conjecture, Ann. of Math. (2), 178 (2013), 321–384.
  • Y. Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, 106, Cambridge University Press, Cambridge, 1993.
  • M. Kiyota, On $3$-blocks with an elementary abelian defect group of order $9$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1984), 33–58.
  • H. Kurzweil and B. Stellmacher, The theory of finite groups, Universitext, Springer-Verlag, New York, 2004.
  • B. Külshammer and T. Okuyama, On centrally controlled blocks of finite groups, unpublished.
  • M. Linckelmann, Introduction to fusion systems, In: Group representation theory, 79–113, EPFL Press, Lausanne, 2007. Revised version:
  • L. J. Mordell, The representation of a definite quadratic form as a sum of two others, Ann. of Math. (2), 38 (1937), 751–757.
  • H. Nagao and Y. Tsushima, Representations of finite groups, Academic Press Inc., Boston, MA, 1989.
  • J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Springer-Verlag, Berlin, 1999.
  • J. B. Olsson, On $2$-blocks with quaternion and quasidihedral defect groups, J. Algebra, 36 (1975), 212–241.
  • W. Plesken, Solving $XX^\textnormal{tr}=A$ over the integers, Linear Algebra Appl., 226/228 (1995), 331–344.
  • L. Puig and Y. Usami, Perfect isometries for blocks with abelian defect groups and cyclic inertial quotients of order $4$, J. Algebra, 172 (1995), 205–213.
  • G. R. Robinson, On the focal defect group of a block, characters of height zero, and lower defect group multiplicities, J. Algebra, 320 (2008), 2624–2628.
  • B. Sambale, $2$-Blocks with minimal nonabelian defect groups, J. Algebra, 337 (2011), 261–284.
  • B. Sambale, Cartan matrices and Brauer's $k(B)$-conjecture II, J. Algebra, 337 (2011), 345–362.
  • B. Sambale, Blocks of finite groups and their invariants, Springer Lecture Notes in Math., 2127, Springer-Verlag, Berlin, 2014.
  • B. Sambale, On the Brauer–Feit bound for abelian defect groups, Math. Z., 276 (2014), 785–797.
  • B. Sambale, Cartan matrices and Brauer's $k(B)$-conjecture III, Manuscripta Math., 146 (2015), 505–518.
  • H. Sasaki, The mod $p$ cohomology algebras of finite groups with metacyclic Sylow $p$-subgroups, J. Algebra, 192 (1997), 713–733.
  • M. Sawabe and A. Watanabe, On the principal blocks of finite groups with abelian Sylow $p$-subgroups, J. Algebra, 237 (2001), 719–734.
  • R. Stancu, Control of fusion in fusion systems, J. Algebra Appl., 5 (2006), 817–837.
  • D. A. Suprunenko, Matrix groups, Amer. Math. Soc., Providence, R.I., 1976.
  • A. Turull, Fixed point free action with regular orbits, J. Reine Angew. Math., 371 (1986), 67–91.
  • Y. Usami, On $p$-blocks with abelian defect groups and inertial index $2$ or $3$. I, J. Algebra, 119 (1988), 123–146.
  • A. Watanabe, Note on a $p$-block of a finite group with abelian defect group, Osaka J. Math., 26 (1989), 829–836.
  • A. Watanabe, Notes on $p$-blocks of characters of finite groups, J. Algebra, 136 (1991), 109–116.
  • A. Watanabe, On perfect isometries for blocks with abelian defect groups and cyclic hyperfocal subgroups, Kumamoto J. Math., 18 (2005), 85–92.
  • A. Watanabe, Appendix on blocks with elementary abelian defect group of order 9, In: Representation Theory of Finite Groups and Algebras, and Related Topics (Kyoto, 2008), 9–17, Kyoto University Research Institute for Mathematical Sciences, Kyoto, 2010.
  • A. Watanabe, The number of irreducible Brauer characters in a $p$-block of a finite group with cyclic hyperfocal subgroup, J. Algebra, 416 (2014), 167–183.
  • S. Yang and S. Gao, On the control of fusion in the local category for the $p$-block with a minimal nonabelian defect group, Sci. China Math., 54 (2011), 325–340.