Groups with exactly one irreducible character of degree divisible by \(p\). (Q2509405)

Let \(p\) be a prime and \(G\) a finite group. The well-known Itô-Michler Theorem states that if no irreducible complex character degree of \(G\) is divisible by \(p\), then \(G\) has a normal Abelian Sylow \(p\)-subgroup. Since then there have been many other results showing that restrictions on the \(p\)-parts of the degrees of the characters influence the \(p\)-structure of the group. In the paper under review, the authors are able to characterize all finite groups which have precisely one irreducible character of degree divisible by \(p\). The classification is a little too long to state here, but as a corollary, the authors prove that if \(P\) is a Sylow \(p\)-subgroup of such a group, then either \(N_G(P)=G\) or \(N_G(P)\) is maximal in \(G\). Not surprisingly, the proof of the main result depends on CFSG.