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By Flaass D.G.

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F) WehaveK",(P);2 Cp(K",(P)) 2 Z(P)andK"'(P);2 C p (K"'(P));2 Z(P). Proof Part (a) follows from the definitions. We prove parts (b), (c), and (d) together. Suppose that Q c:;:; P and Q contains KlP) for all i 2 O. 1) KlP) c:;:; Ki(Q) c:;:; K i+2 (P) for all even i 2 O. 2) by induction on i. 1) is obvious for i = - I. We give the argument from ito i+ 1 for odd i; the proof for even i is similar. (P; KlQ)). :/{'i+ I(P), then A c:;:; K i+I(P) c:;:; Q. (Q; KlQ)) = :/{'i+I(Q). 2) with i replaced by i+ 1 (for our particular choice of i).

Some functors satisfy our conditions and some do not; we must choose particular functors and work with them. This will require further investigation into the structure and embedding of p-subgroups in G. We must also choose the properties we wish to prove. 6, a conjugacy functor that controls strong fusion must control transfer; the converse is false. Thus in some cases we will try to prove the easier result. Finally, we will see that some theorems are true for certain primes, but not for others.

1, and N(S)/C(S) is a p-group, then G has For the following result we require a theorem of Burnside that asserts that aB groups of order paqb are solvable (for a, b ;;::: and distinct primes p, q). This result is proved by character theory (Hall, 1959). Part (b) of the following result proves a conjectHre in Huppert (1967). After (b) was presented at the Conference, Professor B. Klaiber raised the question of whether (a) was valid. 6. Suppose that N(Q) = Q for every prime divisor q of and every Sylow q-subgroup Q of G.

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