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What are the applications of theory of fusion systems to finite group theory or representation theory of finite groups? More concretely, is there any important result in finite group theory or representation theory of finite groups whose prove uses fusion systems in the essential way?

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What is a fusion system? –  Victor Protsak Aug 16 '10 at 2:50
    
Roughly a fusion system on a p-group is a category whose objects are its subgroups and morphisms between them are injective group homomorphism satisfying some axioms. This is supposed to be a generalisation of the case when this p-group is a Sylow subgroup of some group and morphisms are given by conjugations. –  zamanjan Aug 16 '10 at 3:42
    
It actually had two parents, Puig and BLO. For Puig it was a common generalization of the way a finite group conjugates the subgroups of one of its Sylow p-subgroups and the way it conjugates the Brauer subpairs of the defect group of one of its p-blocks. For BLO it is the bare essentials for understanding a (p-completed) classifying space; it didn't quite define the homotopy of the space, but did define its cohomology. The extra glue is called a centric linking system, and the glued result is called a p-local group (but is just a topological space that is poorly understood). –  Jack Schmidt Aug 16 '10 at 3:50
    
So zamanjan's question is very good: They were designed to understand modular representation theory and group cohomology, but what good are they? –  Jack Schmidt Aug 16 '10 at 3:53
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Jack Schmidt has an excellent answer below, so my comment might be moot, but I think the question could use improving. As it is, it is consistent with the possibility that OP saw some words on the internet and wrote a generic question about them. This is probably not what happened, but the question could include OP's answer above to Victor Protsak's comment, and other commentary, explaining why the interest in the question and what OP already knows. Also, please edit the title into a detailed question --- you have 240 characters --- it helps google find it. –  Theo Johnson-Freyd Aug 16 '10 at 4:28
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2 Answers 2

up vote 9 down vote accepted

Lots of results in group cohomology only have topological proofs using the techniques of Bob Oliver and his (generalized) collaborators. For instance, many results along the lines of "controls fusion iff controls cohomology" only have topological proofs using the same techniques that Bob Oliver called fusion systems (though I think some of the papers stick to the topological language). "Controls fusion" is a very old term predating fusion systems by 50 years, so I think of this as a pretty pure finite group cohomology result.

I think it is common to call these topological results by the name "fusion" results, even if fusion systems are not used directly, but rather p-completed classifying spaces are used (also known as p-local groups and such), even if the fusion system used to define the topological space is not prominent. Allowing such abuse of language, one of my favorite examples is the topological proof that the famous Z*-theorem holds for odd primes too: MR1125010. It was proven later using more pure finite group theory, but I suspect lots of people prefer the topological proof.

In modular representation theory, the work of Puig on nilpotent blocks is (as far as I know) literally defined in terms of fusion systems and gets results on the representation theory using only conditions on the fusion systems. One also has some nice descriptions of the representation type of blocks of dihedral defect in terms of the fusion system induced on the block (I believe Linckelmann's introduction gives this as an example).

More historically, the classification of finite groups with dihedral (and then for semi-dihedral/wreathed) begins by dividing clearly into cases based on which fusion system occurs. This just organizes the (couple hundred page) papers, so I guess whether it is essential depends on how practical you are.

Perhaps the most "exciting" answer is not quite ready yet, but Michael Aschbacher and collaborators have begun to recast the classification of finite simple groups in terms of fusion systems. It is quite surprising and comforting that several very important steps in the classification are radically easier in the fusion system case. The fusion system theorems would also have as side-effects corresponding theorems in modular representation theory and the theory of p-local groups.

Let me know if you want references for any of the wishy washy terms, but basically read Linckelmann's introduction, read a few papers of BLO (Broto–Levi–Oliver), and check out the slides from any recent finite group theory conference, and you should have some pretty convincing evidence that fusion systems are opening up a very bright future for finite group theory.

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Thanks Jack, this is exactly the kind of answer I was looking for. –  zamanjan Aug 16 '10 at 16:12
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I think a fair amount of the work of Bob Oliver and his collaborators fits this description. You might start by looking at:

MR2203209 (2007c:55014) Oliver, Bob Equivalences of classifying spaces completed at the prime two. (English summary) Mem. Amer. Math. Soc. 180 (2006), no. 848, vi+102 pp.

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But first read Jack's excellent answer above! –  Dan Ramras Aug 19 '10 at 5:34
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