Alperin's fusion theorem in terms of well-placed tame intersections

Statement
Suppose $$G$$ is a finite group, $$p$$ is a prime, and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Then, the collection of tame intersections involving $$P$$ form a conjugation family for $$P$$ in $$G$$.

Explicit statement using the right-action convention
$$G$$ is a finite group, $$p$$ is a prime, and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Suppose $$A,B$$ are subsets of $$P$$ that are conjugate by some element $$g \in G$$. Then, there exists a collection of fact about::well-placed tame Sylow intersections $$P \cap Q_i, 1 \le i \le n$$ and a collection of elements $$g_i \in N_G(P \cap Q_i)$$ such that:


 * $$\langle A \rangle \le P \cap Q_1$$.
 * $$A^{g_1g_2 \dots g_r} \in P \cap Q_{r+1}$$ for any $$1 \le r \le n - 1$$.
 * $$g = g_1g_2 \dots g_n$$.

Other versions
There are two weaker versions of Alperin's fusion theorem:


 * Alperin's fusion theorem in terms of tame intersections
 * Alperin's fusion theorem in terms of conjugation families

Applications

 * Control of fusion is local: This version of Alperin's fusion theorem is crucial to showing that if a conjugacy functor controls fusion in the normalizers of all non-identity $$p$$-subgroups of a finite group, it controls fusion in the whole group.