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

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) ≥ 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) ≥ p − 1 or else p = 11 and G/{O₏^ } (G) C₁₁²\, \, 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 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 p - 1 + 1 - kₚ (G) 2 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.