A reduction theorem for fusion systems of blocks

Let p be a prime number. Fusion systems on finite p-groups were introduced by L. Puig and provide an axiomatic framework for studying p-fusion in finite groups. This axiomatic point of view has been very useful in determining many properties of finite groups and of the p-completion of their classifying spaces as well as in modular representation theory. As well, it underlies the theory of p-local finite groups developed by C. Broto, R. Levi and R. Oliver.

Let k be an algebraically closed field of characteristic p and G a finite group. An interesting question for fusion systems is whether they can be obtained from the local structure of a block of the group algebra kG. In this talk I present a joint work with Radha Kessar on some methods to reduce this question to the case when G is a central p′-extension of a simple group. As an application of our result, we obtain that the ’exotic’ examples of fusion systems discovered by Ruiz and Viruel do not occur as fusion systems of p-blocks of finite groups. (PDF file)