By Ol'shanskii A. Y.

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The K-subgroups Section 11 illustrated some cases in which conjugacy functors are inadequate. In this section and the following section, we will prove the existence of a section conjugacy functor on G that enjoys the following properties: (i) W(P) is a characteristic subgroup of P and W(P) :2 Z(P), for every section P of G that is a p-group; (ii) if p 2:: 5, then W controls transfer in G; (iii) if p is odd, then W controls weak closure of elements in G; and (iv) if p is odd and every section of G is p-stable, then W controls strong fusion in G.

Let W = W(S). By the definition of a section conjugacy functor, W(SKjK) = WKjK. Clearly, W is a Sylow p-subgroup of WK and N(W) £; N(WK). By the Frattini argument, N(WK) = N(W)(WK) = N(W)K. Hence N(WK)jK has a normal p-complement. However, N(WK)jK = NG/K(WKjK) = NG/K(W(SKjK)). By induction, GjK has a normal p-complement, say, MjK. 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.

Let T = DiG). Suppose that Koo(S) or Koo(S) is not a normal sub-group of G. Then there exists an element 9 of S - T that has the following properties: (a) [X, g; 4] ~ Y for every chieffactor XI Y ofG such that X (b) [Z(T), g, g] = 1; and (c) if C(T) ~ T; = T, then some factor group of G is not p-stable. 1, there exists A E f2(T). 2. 2 yields that [Z(T), g, g] ~ [Z(A), g, gJ = 1. 2. 36 G. GLAUBERMAN For some of the following results, we regard K ro as a section conjugacy functor on G. Note that all of the results below about Kro are valid for Kro as well.

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