Conjugacy functor whose normalizer generates whole group with p'-core controls fusion

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Statement

Suppose G is a finite group, p is a prime number, and W is a p-conjugacy functor on G whose normalizer, along with the p'-core, generates the whole group. Explicitly, this means that if P is a p-Sylow subgroup of G:

\! G = O_{p'}(G)N_G(W(P)).

Then, W controls fusion in G. In other words, any two subsets of P that are conjugate in G are also conjugate in N_G(W(P)).

References