Local Analysis for the Odd Order Theorem by Helmut Bender

By Helmut Bender

In 1963 Walter Feit and John G. Thompson released an explanation of a 1911 conjecture through Burnside that each finite crew of strange order is solvable. This evidence, which ran for 255 pages, used to be a tour-de-force of arithmetic and encouraged extreme attempt to categorise finite uncomplicated teams. This ebook offers a revision and enlargement of the 1st 1/2 the evidence of the Feit-Thompson theorem. less complicated, extra distinct proofs are supplied for a few intermediate theorems. fresh effects are used to shorten different proofs.

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3), CK{R) = 1. 1, G is a Frobenius group with kernel K and complement R. 3 implies that Cy(R) # 0, which contradicts our original assumption. Thus K is a nonabelian special group. 3), R centralizes Z(K). Thus Z(K) C Z(G), and hence Z(K) is cyclic. Consequently K is extraspecial. 3), R acts irreducibly on K/K1 = K/Z(K) and Z(K) is a maximal ^-invariant subgroup of K. Thus Z(K) = CK{R) = CK(x) for all x G R*. 5 to obtain a final contradiction. 5. Let G = KR be a Frobenius group with solvable Frobenius kernel K and cyclic Frobenius complement R of prime order.

Ytl for each i = 1,2,3, Since 5 = (y,T), we have ^7 = 7/? and (3P = 1. Thus 5 is an elementary abelian p-group. Since CR(T)/C is isomorphic to a subgroup of J5, CR(T)/C is an elementary abelian p-group. 20), TC/C has an A-invariant complement X/C in CR(T)/C. 5, X is cyclic. Let X = (x). 12), # = SCR(T) = SX. Since both S and T centralize S/Sf, we know that X does not centralize 5/5". That is, [X,S\£Sf. 15) Now \S/T\ = \T/Sf\ = p. Therefore, by taking any elements y e 5 -T and z e T - 5', we get S/T={yT) and T/S' = (zS').

2 now yield [y,zY = [y,z]tt = [i/ t t ,^] = [yj,zk] = [y,Zyk, and [x,y]k = [x,y)« = [x a ,j/ a ] = [x*,^] = [x,yp (mod S'). , jA: = i (mod p), ij = k (mod p), i j 2 = i (mod p), and j 2 = 1 (mod p), a contradiction. 17. Suppose p is an odd prime, R is a p-group, and A is a solvable group of automorphisms of R. Assume that T(R) < 2 and \A\ is odd. Then A' is a p-group. Proof. We may assume that R ^ 1. 13. 16) CA{H) is a p-group. Furthermore, if has exponent p and r(if) < T(R) < 2. 17) \H\

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