Primes dividing the degrees of the real characters.
Let \(G\) be a finite group and \(P\in\text{Syl}(G)\). The main result of the paper proves that if all real-valued irreducible characters of \(G\) have odd degree, then \(P\) is normal in \(G\). This is a version of the Ito-Michler Theorem for real-valued characters and the prime 2. Its proof depends on CFSG, and the authors remark that it does not remain true for odd primes, but defer the study of this case to a later paper. The second main result of the paper studies a graph whose vertices are the primes dividing the degrees of the real-valued irreducible character degrees, and two such primes are joined by an edge if their product divides the degree of some real-valued irreducible character degree. They show that this graph has at most three connected components, and if \(G\) is solvable, it has at most two connected components. The paper also contains some results on similar questions for the real conjugacy class sizes of a group; the proofs here are quite a bit easier.
scientific article; zbMATH DE number 5294687
scientific article