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

Let k(B0) and l(B0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B0 of a finite group G. We prove that, if k(B0)− l(B0) = 1, then l(B0) ≥ p− 1 or else p = 11 and l(B0) = 9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B0) ≥ p− 1 or else p = 11 and G/Op′(G) ∼= C 11 ⋊ 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 √ p− 1+ 1− kp(G), where kp(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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