Alperin's fusion theorem in terms of well-placed 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 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 Well-placed 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} \in P \cap Q_{r+1} for any 1 \le r \le n - 1.
  • g = g_1g_2 \dots g_n.

Related facts

Other versions

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

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.

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 284, Theorem 4.5, Chapter 8 (p-constrained and p-stable groups), Section 4 (Groups with subgroups of glauberman type), More info