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 \(G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)\) $$G/O_{p^{\prime }}(G)\cong C_{11}^2⋊\text{SL}(2,5)$$ . 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 \(2\sqrt {p - 1} + 1 - {k_p}(G)\) $$2\sqrt{p-1}+1-k_p(G)$$ , 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.