Alperin's fusion theorem in terms of tame intersections

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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.

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