Group in which the ZJ-functor controls fusion for a prime

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Suppose G is a finite group and p is a prime number. We say that G is a group in which the ZJ-functor controls fusion for p if the ZJ-functor, viewed as a conjugacy functor on G, controls fusion in G with respect to p. In other words, given a p-Sylow subgroup P of G and two subsets A,B of P that are conjugate in G, the subsets A,B are conjugate in N_G(Z(J(P)).

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