On saturated fusion systems and Brauer indecomposability of Scott modules

Let p be a prime number, G a finite group, P a p-subgroup of G and k an algebraically closed field of characteristic p. We study the relationship between the category FP(G) and the behavior of p-permutation kG-modules with vertex P under the Brauer construction. We give a sufficient condition for FP(G) to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we give a criterion for the categories arising from the data of (b,G)-Brauer pairs in the sense of Alperin–Broué and Broué–Puig to be saturated fusion systems on the underlying p-group.