Zeros of real irreducible characters of finite groups

Abstract

We prove that if all real-valued irreducible characters of a finite group $G$ with Frobenius–Schur indicator $1$ are nonzero at all $2$-elements of $G$, then $G$ has a normal Sylow $2$-subgroup. This result generalizes the celebrated Ito–Michler theorem (for the prime $2$ and real, absolutely irreducible, representations), as well as several recent results on nonvanishing elements of finite groups.