Primes dividing the degrees of the real characters

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ ∈ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irr_{rv} (G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr_{rv} (G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.