Abstract
Let G be a finite group, let N be a normal subgroup of G and let θ be an irreducible character of N. P.X. Gallagher showed that the number of irreducible characters of G lying θ equals the number of so-called θ-good conjugacy classes of Gθ.Here we count the number of real irreducible characters of G lying over θ. To do so, we assign a new invariant, with value 0,+1 or −1, to each good conjugacy class. Then the desired number is the sum over these invariants.We also compute the Frobenius-Schur indicator of the induced character θ↑G using a similar formula.
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