On almost p-rational characters of p ′ p^{\prime} -degree

Abstract

Abstract Let p be a prime and let G be a finite group. A complex character of G is called almost p-rational if its values belong to a cyclotomic field ℚ ⁢ ( e 2 ⁢ π ⁢ i / n ) {{\mathbb{Q}}(e^{2\pi i/n})} for some n ∈ ℤ + {n\in{\mathbb{Z}}^{+}} not divisible by p 2 {p^{2}} . We prove that, in contrast to usual p-rational characters, there are “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 ′ {p^{\prime}} -degree and prove that, in the minimal situation, the number of almost p-rational irreducible p ′ {p^{\prime}} -characters of G coincides with that of 𝐍 G ⁢ ( P ) {{\mathbf{N}}_{G}(P)} for P ∈ Syl p ⁡ ( G ) {P\in{\operatorname{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 ′ {p^{\prime}} -characters and the blockwise refinement of the McKay–Navarro conjecture.