finite groups - Applications of fusion systems

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.