TY - JOUR

T1 - Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

AU - Hung, Nguyen Ngoc

AU - Sambale, Benjamin

AU - Tiep, Pham Huu

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/9

Y1 - 2024/9

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.

KW - math.RT

KW - math.GR

UR - http://www.scopus.com/inward/record.url?scp=85191687731&partnerID=8YFLogxK

U2 - 10.1007/s11856-024-2613-1

DO - 10.1007/s11856-024-2613-1

M3 - Article

VL - 262

SP - 327

EP - 358

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -