Variations on McKay's character degree conjecture (Q1814054)

All groups considered are finite and \(p\) is prime. If we let \(t(G)\) denote the number of irreducible characters of \(G\) of \(p'\)-degree and let \(P\in \text{Syl}_ p(G)\), then McKay's conjecture proposes that \(t(G)= t(N_ G(P))\). This has been proven for \(p\)-solvable \(G\) by Dade and also by Okuyama and Wajima. The analogous result for irreducible Brauer characters (relative to the same prime \(p\)) is known to be true for \(p\)- solvable \(G\), but also known to be false in general. Our results here generalize this in two directions. First, if \(q\) is also prime, we show that the number of irreducible Brauer characters (for the prime \(p\)) of \(q'\)-degree is the same for \(G\) as for the normalizer of a Sylow-\(q\)- subgroup of \(G\), provided \(G\) is \(\{p,q\}\)-solvable. We also replace \(p\) and \(q\) by sets of primes.