Characters of p'-degree in solvable groups
The groups considered are finite and the group characters are defined over the complex numbers. For a group \(G\) and a prime \(p\), we let \(\text{Irr}(G)\) be the set of irreducible characters of \(G\) and let \(I_p(G) = \{\psi \in \text{Irr}(G)|(p, \psi(1)) = 1\}\). The McKay conjecture proposes \(|I_p(G)| =|I_p(N(P))|\) where \(P \in \text{Syl}_p(G)\). Originally, the conjecture was for simple \(G\) and \(p=2\). \textit{I. M. Isaacs} has proven the result when \(|G|\) is odd and \(p\) is any prime. This paper proves the result for solvable \(G\) and any prime \(p\). For solvable \(G\), this is generalized to sets of primes and normalizers of Hall subgroups.
scientific article; zbMATH DE number 3558043