A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters

Abstract Let 𝐺 be a finite group and let Irr s ⁢ ( G ) \mathrm{Irr}_{\mathfrak{s}}(G) be the set of irreducible complex characters 𝜒 of 𝐺 such that χ ⁢ ( 1 ) 2 \chi(1)^{2} does not divide the index of the kernel of 𝜒. In this paper, we classify the finite groups 𝐺 for which any two characters in Irr s ⁢ ( G ) \mathrm{Irr}_{\mathfrak{s}}(G) are Galois conjugate. In particular, we show that such groups are solvable with Fitting height 2.