Real ordinary characters and real Brauer characters

Abstract

We prove that if G G is a finite group and p p is a prime such that the degree of every real-valued irreducible complex, respectively real-valued irreducible p p -Brauer character, of G G is coprime to p p , then O p ′ ( G ) \mathbf {O}^{p’}(G) is solvable. This result is a generalization of the celebrated Ito–Michler theorem for real ordinary characters, respectively real Brauer characters, with Frobenius-Schur indicator 1 1 .