Alperin's fusion theorem in terms of conjugation families
Contents
History
This statement was formulated and proved by Jonathan Lazare Alperin, in 1965.
Statement
General form
Suppose is a finite group, is a -Sylow subgroup for a prime Suppose is a collection of subgroups of with the property that for any subgroup of , there exists such that:
- and are conjugate subgroups inside
- is a -Sylow subgroup of
Then is a conjugation family for in .
More specific form
This form states:
Let be the collection of all subgroups of whose normalizer in is a Sylow subgroup of the normalizer in . Then, is a conjugation family for in .
This follows from the more general form, and the fact that every p-subgroup is conjugate to a p-subgroup whose normalizer in the Sylow is Sylow in its normalizer.
Proof
Key idea: induction on index
We prove the result by inducting on the index of the subgroup in . Suppose and .
Base case for induction
The base case of induction is when . By the conditions, contains a conjugate of , so . Clearly, then , so we can set , and .
Induction step
The key thing to remember for the induction step is that if is a proper subgroup of , then is a proper subgroup of (and similarly for ). Thus, this step reduces to three parts:
- Go from to
- Use the induction to argue that we can go from to
- Go from back down to
References
Journal references
- Transfer and fusion in finite groups by Jonathan Lazare Alperin and Daniel Gorenstein, Journal of Algebra, ISSN 00218693, Volume 6, Page 242 - 255(Year 1967): This paper discusses the normalizers of subgroups of a Sylow subgroup in a finite group, using the ideas of a conjugation family and Alperin's fusion theorem^{Weblink (hosted on ScienceDirect)}^{More info}
Textbook references
- Book:GL^{More info}, Page 6-7, Theorems 3.4 and 3.5