Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

Research output: Working paper/Preprint › Preprint

Authors

  • Nguyen Ngoc Hung
  • Benjamin Sambale
  • Pham Huu Tiep

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Original languageEnglish
Publication statusE-pub ahead of print - 8 Feb 2021

Abstract

Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least 2\sqrt{p-1}+1-k_p(G), where k_p(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

Keywords

    math.RT, math.GR

Cite this

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements. / Hung, Nguyen Ngoc; Sambale, Benjamin; Tiep, Pham Huu.
2021.

Research output: Working paper/Preprint › Preprint

Hung, N. N., Sambale, B., & Tiep, P. H. (2021). Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements. Advance online publication. http://arxiv.org/abs/2102.04443v2
Hung NN, Sambale B, Tiep PH. Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements. 2021 Feb 8. Epub 2021 Feb 8.
Hung, Nguyen Ngoc ; Sambale, Benjamin ; Tiep, Pham Huu. / Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements. 2021.
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N2 - Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least 2\sqrt{p-1}+1-k_p(G), where k_p(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

AB - Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least 2\sqrt{p-1}+1-k_p(G), where k_p(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

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