Characterizing normal Sylow \(p\)-subgroups by character degrees. (Q1950641)

Let \(G\) be a finite group, \(p\) a prime number and \(P\) a Sylow \(p\)-subgroup of \(G\). The well-known Itô-Michler theorem states that \(P\) is Abelian and normal in \(G\) if and only if every irreducible character of \(G\) has degree not divisible by \(p\). In this paper the authors answer the question whether some canonical subset of the irreducible characters of \(G\) exactly captures the normality of \(P\) in the affirmative as follows: \(P\) is normal in \(G\) if and only if \(p\) does not divide the degree of any irreducible constituent of the permutation character \((1_P)^G\). They also show that this condition is equivalent to the property that \(\chi(x)\neq 0\) for all \(x\in P\) and all irreducible constituents \(\chi\) of the permutation character \((1_P)^G\). The proofs depend on CFSG.