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Exotic fusion systems and pearls Grazian, Valentina


Fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for p any prime number. A fusion system F on a p-group S is a category where the objects are all the subgroups of S and the morphisms are certain injective morphisms that â behave as conjugationsâ (in particular all restrictions of conjugation maps induced by elements of S are morphisms in F). Given a finite group G, it is always possible to define the saturated fusion system realized by G on one of its Sylow p-subgroups S: this is the category where the morphisms are the restrictions of conjugation maps induced by the elements of G. However, not all saturated fusion systems can be realized in this way. When this is the case, we say that the fusion system is exotic. The understanding of the behavior of exotic fusion systems (in particular at odd primes) is still an important open problem. In this talk we aim to present a new approach to the study of exotic fusion systems at odd primes. First, we will introduce the notion of pearls: essential subgroups that are either elementary abelian of order p^2 or non-abelian of order p^3. Then we will show the connections between pearls and exotic fusion systems and we will present new results concerning the classification of saturated fusion systems containing pearls.

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