Alperin's fusion theorem in terms of well-placed tame intersections
Explicit statement using the right-action convention
is a finite group, is a prime, and is a -Sylow subgroup of . Suppose are subsets of that are conjugate by some element . Then, there exists a collection of Well-placed tame Sylow intersection (?)s and a collection of elements such that:
- for any .
There are two weaker versions of Alperin's fusion theorem:
- Alperin's fusion theorem in terms of tame intersections
- Alperin's fusion theorem in terms of conjugation families
- 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 -subgroups of a finite group, it controls fusion in the whole group.