Abstract
We introduce objective partial groups, of which the linking systems and p-local finite groups of Broto, Levi, and Oliver, the transporter systems of Oliver and Ventura, and the \({\mathcal{F}}\)-localities of Puig are examples, as are groups in the ordinary sense. As an application we show that if \({\mathcal{F}}\) is a saturated fusion system over a finite p-group then there exists a centric linking system \({\mathcal{L}}\) having \({\mathcal{F}}\) as its fusion system, and that \({\mathcal{L}}\) is unique up to isomorphism. The proof relies on the classification of the finite simple groups in an indirect and—for that reason—perhaps ultimately removable way.
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Chermak, A. Fusion systems and localities. Acta Math 211, 47–139 (2013). https://doi.org/10.1007/s11511-013-0099-5
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DOI: https://doi.org/10.1007/s11511-013-0099-5