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

https://doi.org/10.1007/s11856-024-2613-1

Hung, Nguyen Ngoc; Sambale, Benjamin; Tiep, Pham Huu (September 2024, Israel Journal of Mathematics)

Abstract Letk(B₀) andl(B₀) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB₀of a finite groupG. We prove that, ifk(B₀)−l(B₀) = 1, thenl(B₀) ≥p− 1 or elsep= 11 andl(B₀) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B₀) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$

G / O_{p^{'}} (G) ≅ C_{11}^{2} ⋊ SL (2, 5)

. These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$

2 \sqrt{p - 1} + 1 - k_{p} (G)

, wherek_p(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups.

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