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By Pannenberg M.

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13. 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

Thus i is odd. ObvioU61y, Ki-iS) S; G = N(A). 7) [A, A*] \$; Z(A). 8) A*-T. 7), 9 satisfies (b) and S; A (\ [A*, g] c:; A (\ (A*)' S; Z(A). This proves (a). 3. Thus we may assume that A E Jf'i-l(S), Take BE Jf'i(S) such that B \$; T. Take 9 E B-T. Suppose first that i is odd. Then [B, K i - 1(S)] S; Z(B), and consequently [A, B, B] = 1. This proves (a) and (b). 3, we obtain (c). Now assume that i is even. Then A normalizes B. Hence [B, A] S; [[A, B], B, B] S; [B, B, B] S; [Z(B), B] = 1, which yields (a).

4. If P is odd, then J controls weak closure of elements in G. 5. Let Z = Z(sy. Suppose A is an element of Z 11 Z(N(J(S»), or A is a normal subgroup ofN(J(S» contained in Z. Then A is weakly closed in S with respect to G. GLOBAL AND LOCAL PROPERTIES OF FINITE GROUPS 43 Suppose p is odd. 2, Gorenstein, 1968) asserts that G is p-stable if SL(2, p) is not involved in G. 6 and the definition of p-stability, we may state their theorem as follows. 6. Assume that p is odd. Suppose that, for every non-identity p-subgroup P of G, the group SL(2, p) is not involved in N(P).