Sylow normalizers and Brauer character degrees
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.
scientific article; zbMATH DE number 1502167