Fusion systems, linking systems and punctured groups

Abstract: Saturated fusion systems and associated linking systems are categories modelling the $p$-local structure of finite groups. In particular, linking systems contain the algebraic information that is needed to study $p$-completed classifying spaces of fusion systems similarly to $p$-completed classifying spaces of finite groups. If $G$ is a finite group and $S$ is a Sylow $p$-subgroup of $G$, then we can construct a saturated fusion system $\F_S(G)$ as follows: The objects are all subgroups of $S$, and the morphisms between two objects are the injective group homomorphisms induced by conjugation with elements of $G$. Saturated fusion systems which do not arise in this way are called exotic.

The concept of a linking system was generalized by Oliver and Ventura to transporter systems. Andrew Chermak introduced moreover group-like structures, called localities, which correspond in a certain way to transporter systems. I will give an introduction to the subject and outline how the theory of localities can be used to prove new theorems on fusion systems. Moreover, I will report on a project with Assaf Libman and Justin Lynd, where we study "punctured groups''. Here a transporter system (or a locality) associated to fusion system $\F$ over $S$ is called a punctured group if the object set is the collection of all non-identity subgroups. It should be noted in this context that a fusion system $\F$ over a $p$-group $S$ can be realized as a category $\F_S(G)$ as above if and only if there is a transporter system whose object set is the full collection of subgroups of $S$. In particular, to every group fusion system one can associate a punctured group. In the project with Libman and Lynd, we determine for many of the known exotic fusion systems whether an associated punctured group exists.

algebraic topologycategory theorygroup theoryK-theory and homology

Audience: researchers in the topic

Cihan Okay

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