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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Frobenius-Schur indicator
- Nonvanishing element
- Real irreducible character