\(p\)-singular characters and normal Sylow \(p\)-subgroups (Q6545242)

Let \(G\) be a finite group and \(p\) a prime. Write \(\mathrm{Irr}_p(G)\) for the set of irreducible complex characters of \(G\) whose degrees are linear or divisible by \(p\) and write \(\mathrm{R}_p(G)\) for the ratio of the sum of squares of irreducible character degrees in \(\mathrm{Irr}_p(G)\) to the sum of irreducible character degrees in \(\mathrm{Irr}_p(G)\). The Itô-Michler theorem on character degrees states that \(\mathrm{R}_p(G)=1\) (which is equivalent to saying that \(p\) does not divide any irreducible character degree of \(G\)) if and only if \(G\) has a normal abelian Sylow \(p\)-subgroup. \N\NThe main result of the paper under review is the following: if \(\mathrm{R}_p(G)<\frac{p+1}{2}\), then \(G\) has a normal Sylow \(p\)-subgroup. \N\NThis extends other work by some researchers who studied \(R(G)\), which is \(R_p(G)\) with all the \(p\)'s removed or in other words, the quotient of \(|G|\) and the sum of the degrees of all irreducible complex characters. For example, \textit{H. P. Tong-Viet} [Arch. Math. 99, No. 5, 401--405 (2012; Zbl 1259.20009)] showed that if \(R(G) < 15/4\), then \(G\) is solvable and this bound is sharp. Also, \textit{A. Maróti} and \textit{H. N. Nguyen} [Forum Math. 27, No. 4, 2453--2465 (2015; Zbl 1329.20008)] proved that if \(R(G) \leq p/\sqrt{3}\) for a prime \(p\), then \(G\) is \(p\)-solvable. Thus studying \(R_p(G)\) is a natural next step in this context.