Sylow normalizers and Brauer character degrees (Q1579158)

Let \(p\) and \(q\) be primes and let \(G\) be a finite \(\{p,q\}\)-solvable group. Let \(P\) be a Sylow \(p\)-subgroup of \(G\) and \(Q\) a Sylow \(q\)-subgroup of \(G\). Let \(N_G(P)\) and \(N_G(Q)\) denote the normalizers in \(G\) of these subgroups. The main result of the paper under review is the following. Each irreducible \(p\)-modular Brauer character of \(G\) that has \(q'\)-degree also has \(p'\)-degree if and only if \(N_G(Q)\) is contained in some conjugate of \(N_G(P)\). The authors also give examples to show that their main theorem is not true without some solvability hypothesis. As a corollary, they prove that it is possible to decide from the character table of \(G\) whether or not \(N_G(Q)\) is contained in a conjugate of \(N_G(P)\). The main tool in the proof is Clifford theory, combined with the delicate techniques of analysis of characters of (generalized) solvable groups developed by I. M. Isaacs and G. Navarro.