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By Abrashkin V.A.

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9) yields that [X, g, g] = [(X IIA)Y, g, g] £; [X 11 A, g, g]Y = Y, as claimed. 5. Hence L is a subnormal subgroup of G. 11) if X/ Y is a chief factor of G and X£; A. 2; we obtain (c). 14. The Thompson Subgroup Definitions. Suppose that P is a finite p-group. Let d(P) be the maximum of the orders of the abelian subgroups of P. ) be the subgroup of P generated by the elements of d(P). The group J(P) is called the Thompson subgroup of P. We may regard J and ZJ (given by ZJ(P) = Z(J(P))) as section conjugacy functors on G.

Let H = N(Z(J(S»). Suppose that p is odd and ZJ does not control strong fusion in G. NH(C)/CH(C)I = p-l. 13 is proved by the author in work to be published. 13. The next result is also proved in the same work; however, it was obtained independently by M. J. Collins (1969), with a shorter proof. 14. Suppose that p is odd, T s; Z(S), and T

Then M is a normal p'-subgroup of G and GjM is a p-group. 10, M is a normal p-complement in G, a contradiction. Thus G has no non-identity normal p'-subgroups. 5). Obviously, P is a Sylow p-subgroup of L. 6), CL(P) has no non-identity characteristic p'-subgroups. 4 applied to L instead of G, CL(P) s P. Since CCP)jCL(P) is isomorphic to a 28 G. GLAUBERMAN subgroup of GIL, C(P)/CL(P) is a p-group. Therefore, C(P) is a p-group. Since C(P)

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