# 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 $P \cap Q_i, 1 \le i \le n$ and a collection of elements $g_i \in N_G(P \cap Q_i)$ such that:

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