Further refinements of the Itô-Michler theorem are proved, with the p'-degree condition restricted to p-rational characters lying above the principal character of a Sylow p-subgroup (or strongly real characters when p=2).
L\"ubeck, Elements of order 2 and 3 in exceptional groups of Lie type, https://www.math.rwth-aachen.de/ Frank.Luebeck/chev/23elts.html
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Degrees of $p$-rational characters and normality of Sylow $p$-subgroups
Further refinements of the Itô-Michler theorem are proved, with the p'-degree condition restricted to p-rational characters lying above the principal character of a Sylow p-subgroup (or strongly real characters when p=2).