Abstract: One of the main problems in the theory of fusion systems is the question whether a fusion system arises in the form of a finite group if and only if it arises in the form of a p-block of a finite group. There is a conjecture saying that a fusion system is induced by a group if and only if it is induced by a block. We will present reduction theorems for this question which reduce this problem to blocks of quasisimple groups in certain cases. We will use one of these reductions to settle this question for the family of Parker–Semeraro systems. Finally, we will discuss ongoing work concerning our strategy in order to prove the conjecture for some (simple) groups of Lie type.

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