### Fusion and pearls: what is known

Algebra and Geometry Seminar

31st January 2018, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

In finite group theory, the word fusion refers to the study of conjugacy maps between subgroups of a group. The modern way to solve problems involving fusion is via the theory of fusion systems. A saturated fusion system on a p-group S is a category whose objects are the subgroups of S and whose morphisms are the monomorphisms between subgroups which satisfy certain axioms. The first part of this talk is about the state of the art concerning the classification of simple fusion systems.

To classify saturated fusion systems, we first need to determine the so-called essential subgroups of S. These are self-centralizing subgroups of S whose automorphism group has a restricted structure. We call pearls the essential subgroups of S that are either elementary abelian of order p^2 or non-abelian of order p^3 (and exponent p whenever p is odd). In the second part of this talk we present new results about fusion systems involving pearls and we explain how such results contribute to the classification of simple fusion systems.

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