Alperin's fusion theorem in terms of 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 Sylow 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 Tame Sylow intersection (?)s $P\cap Q_{i},1\leq i\leq n$ and a collection of elements $g_{i}\in N_{G}(P\cap Q_{i})$ such that:

• $\langle A\rangle \leq P\cap Q_{1}$.
• $A^{g_{1}g_{2}\dots g_{r}}\leq P\cap Q_{r+1}$ for any $1\leq r\leq n-1$.
• $g=g_{1}g_{2}\dots g_{n}$.

Related facts

• Alperin's fusion theorem in terms of conjugation families: This is a somewhat weaker version.
• Alperin's fusion theorem in terms of well-placed tame intersections: This is a somewhat stronger version.