# Character Theory for the Odd Order Theorem by T. Peterfalvi, R. Sandling

By T. Peterfalvi, R. Sandling

The well-known theorem of W. Feit and J. G. Thompson states that each team of wierd order is solvable, and the facts of this has approximately components. the 1st seemed in Bender and Glauberman's neighborhood research for the unusual Order Theorem, quantity 188 during this sequence. the current publication presents the character-theoretic moment half and completes the facts. Thomas Peterfalvi additionally bargains a revision of a theorem of Suzuki on break up BN-pairs of rank one, a prerequisite for the type of finite basic teams.

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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 .