The continuity of 𝑝-rationality and a lower bound for 𝑝’-degree characters of finite groups

Let p p be a prime and G G a finite group. We propose a strong bound for the number of p ′ p’ -degree irreducible characters of G G in terms of the commutator factor group of a Sylow p p -subgroup of G G . The bound arises from a recent conjecture of Navarro and Tiep [Forum Math. Pi 9 (2021), pp. 1–28] on fields of character values and a phenomenon called the continuity of p p -rationality level of p ′ p’ -degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture (see G. Navarro [Ann. of Math. (2) 160 (2004), pp. 1129–1140]). We achieve both the bound and the continuity property for p = 2 p=2 .