Bounding \(p\)-Brauer characters in finite groups with two conjugacy classes of \(p\)-elements (Q6635458)

Let \(p\) be a prime and let \(G\) be a finite group with Sylow \(p\)-subgroup \(P\) and \(k_p(G)\) conjugacy classes of \(p\)-elements. Moreover, let \(B_0\) be the principal \(p\)-block of \(G\), let \(l(B_0)\) be the number of irreducible Brauer characters in \(B_0\), and let \(k_0(B_0)\) be the number of irreducible complex characters of height zero in \(B_0\). The authors show: If \(k_p(G) = 2\) then one of the following holds:\N\begin{itemize}\N\item[(i)] \(l(B_0) \geq p\);\N\item[(ii)] \(l(B_0) = p-1\), and \(N_G(P)/O_{p'}(N_G(P))\) is a Frobenius group with a kernel of order \(p\) and a complement of order \(p-1\);\N\item[(iii)] \(p=11\), \(l(B_0) = 9\), and \(G/O_{p'}(G)\) is a Frobenius group with an elementary abelian kernel of order \(121\) and a complement isomorphic to \(\mathrm{SL}(2,5)\).\N\end{itemize}\NThe authors also prove: If \(k_p(G) = 2\) then \(k_0(B_0) \geq p\), or \(p=11\) and \(k_0(B_0) = 10\). In addition, the authors consider the case \(k_p(G) = 3\). Furthermore, they conjecture that \(l(B_0) \geq 2 \sqrt{p-1} + 1 - k_p(G)\) whenever \(k_p(G) > 1\), and they show that this inequality follows from Alperin's weight conjecture.