\(p\)-Brauer characters of \(q\)-defect \(0\) (Q1340663)

Let \(G\) be a finite group, \(p\) a prime number, \(P\) a \(p\)-subgroup of \(G\), and denote by \(\text{IBr}(G,P)\) the set of irreducible \(p\)-Brauer characters with vertex \(P\). If \(q\) is another prime, denote by \(\text{IBr}^ q(G,P)\) the set of characters \(\beta\in\text{IBr}(G,P)\) satisfying \(\beta(1)_ q=| G|_ q\) (called characters of \(q\)- defect zero). The main result of the paper tells that if \(G\) is solvable, then \(|\text{IBr}^ q(G,P)|\leq|\text{IBr}^ q (N_ G(P),P)|\). It is well-known that if \(G\) is \(p\)-solvable, then \(|\text{IBr} (G,P)|=|\text{IBr}(N_ G(P),P)|\), so the above theorem provides additional information in the solvable case. The authors show that the equality does not hold in general, and there is some indication that the theorem may be true for \(p\)-solvable groups.