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Character Quotients for Coprime Acting Groups

Let the finite group A be acting on a finite group G with (∣A∣, ∣G∣)=1. Let Γ be the semidirect product of A and G. Let χ be a character of Γ irreducible after restriction to G. In a previous paper by Brian Hartley and the author, we proved that the restriction of χ to S belongs to the set C(S) obtained by running over all χ that arise in this manner, by assuming, in addition, that G is a product of extraspecial groups. This was proved in general, assuming only some condition on the Green functions of groups of Lie type that is not as yet fully verified. In the present paper, we define the map Q(χ): S↦C by Q(χ)(s)=∣CG(s)∣/χ(s). We prove that Q(χ)∈C(S) under the same hypotheses. In particular, the character quotient Q(χ) is an ordinary character.