# 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 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