By Li J., Su Y., Zhu L.

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W'/j. Suppose that there is an index fi ~ 1 such that fi i- j, fi i- k, III E Sand I1f(l) = I1k(l). 9), 11;;1 -11? = Dk 2::;(wik - wfc)· Since 11? 2). (a) Let S C IrrL be such that Z[S,L#] = Z[S,A] and lSI ~ 2. Let u be an automorphism of the field QIGI for which SU C S. Let 71 be a linear isometry from Z[S] to Z[IrrG] which coincides with 7 on Z[S, A]. Then, if XES, XTIU = XUTI . (b) Let X E Irr L be such that Supp(X) C AU{l}. Then there is a character xY = 11 - p;. Let 1jJ E S, 1jJ i- x. By the definition of 7, 11 E Irr G such that (X - Proof.

M}. (a) If two elements of X are conjugate in G, they are conjugate in M. (b) DC Al(M) and, for xED, Ca(x) is contained in a unique maximal subgroup of G. (c) For xED and L the maximal subgroup ofG for which Ca(x) C L: (el) L = LF >1 (M n L) and Ca(x) = CLF(X) >1 CM(x). (c2) ILFI is prime to ICM(y)1 for all y E x. (c3) x E A(L) - AI(L). (c4) L is of Type I or II. Furthermore, M is a Frobenius group with kernel MF if L is of Type II. Reference. [BG], § 16, Theorem II, Theorem B(5) and Theorem D(4).

Set Y = ax? - 2:;'=1 AiX? + Z with Ai E C and where Z E CF(G) is orthogonal to S{'. For 1 ::; i ::; n, X? e). It follows that, for 1 < i ::; n, aaillXlll 2 Th al aiAlllXlll2 _ ai IIXill 2 - A IIxdl 2' where A = us, Ai = = 1. ) = (-Y, xi' - aiXr' ) = Adlxdl 2+ ai( a - Al)lIxlIl 2. 2 ' _, al . All1xll1 . ) = (a - IIX~1I2) IIxdl 2E Z, o and so A E Z. 2) Y = ax? Proof. We note first that IIxII 2+ a211Xlll2 = lI(x - aXltll2 = IIXII 2 + 11Y1I2. a), IIXII 2 2: IIx1I2, and so 11Y1I2 ::; a211X1112. 1), CI:1I2 - a) 2 11xd2 + A2 ~ 1I::1I4 I1XiIl 2+ IIZII 2S; a211Xl1l2, or Let b = 2a n 2' L~ i=1 .