On almost $p$-rational characters of $p'$-degree

Abstract: Let $p$ be a prime and $G$ a finite group. A complex character of $G$ is called almost $p$-rational if its values belong to a cyclotomic field $\mathbb{Q}(e^{2\pi i/n})$ for some $n\in \mathbb{Z}^+$ prime to $p$ or precisely divisible by $p$. We prove that, in contrast to usual $p$-rational characters, there are always "many" almost $p$-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost $p$-rational irreducible characters of $G$ in terms of $p$. In fact, motivated by the McKay-Navarro conjecture, we obtain the same bound for the number of such characters of $p'$-degree and prove that, in the minimal situation, the number of almost $p$-rational irreducible $p'$-characters of $G$ coincides with that of $N_G(P)$ for $P\in\mathrm{Syl}_p(G)$. Lastly, we propose a new way to detect the cyclicity of Sylow $p$-subgroups of a finite group $G$ from its character table, using almost $p$-rational irreducible $p'$-characters and the blockwise refinement of the McKay-Navarro conjecture.