By Walter Feit

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It suffices to prove the result for every character, and hence for every irreducible character, of the abelian group < P, G >. Let e be an irreducible character of < P, G >. Then 6(1) 1 and 6(PG) = 6 (P)6(G). Furthermore 6 (P) is a pm-th root of unity for some m and so 6 (P) 1(mod ~). The result follows. (Solomon [2]) Let sr l , ... , sr'k be the conjugate classes of ®. Let G . ) is a nonJ - - J= J negative rational integer for any irreducible character X of ®. 5) Proof. Let ! (G) = G-l HG for G, H e::: ®.

The first statement follows directly from the definition of £1*. Let X be an irreducible character of @ and let 9 be an irreducible constituent of X IS)' Then X c £1* by the Frobenius reciprocity theorem. \) has the required form. 10) is the index of ramification of X with respect to Sj. 11) Let X be an irreducible character of (};. Let ~)

Can be found. Since w. (l) is determined. ) 1 J = I@! ). 1 1 J §8. INDUCED REPRESENTATIONS Let ,\) be a subgroup of @. Let ,\)Gu ... ,,\)G m be all the distinct right cosets of ,(,') in @. ,. GG:l)) for G E: @, 1 J where each pair of indices denotes a sub matrix of degree n. f) of degree n then F* is an tr-representation of @ of degree 1@:3)1 n. Proof. For G,H Then iV*(G) ~*(H) E: @ let B .. ). GG-l) i'V(G HG:1 )). GG-tl - 1 E: ~. Thus B.. GG-tl) - IJ 1 lY(GtHG:1 ). H-IG-tl t J 1 J 1 - J 1 and this is the case if and only if GtHG: E: ~.