  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.
