Alperin's fusion theorem in terms of well-placed tame intersections
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Statement
Suppose is a finite group, is a prime, and is a -Sylow subgroup of . Then, the collection of tame intersections involving form a conjugation family for in .
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 .
- .
Related facts
Other versions
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
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 -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}