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

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_p^ (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.