Simple fusion systems over p-groups with unique abelian subgroup of index p

Fix a prime $p$. The fusion system of a finite group $G$ with respect to a Sylow $p$-subgroup $S$ of $G$ is the category $\mathcal{F}_S(G)$ whose objects are the subgroups of $S$, and whose morphisms are the homomorphisms induced by conjugation in $G$. More generally, an abstract fusion system over a $p$-group $S$ is a category $\mathcal{F}$ whose objects are the subgroups of $S$ and whose morphisms are injective homomorphisms between the subgroups that satisfy certain axioms formulated by Lluis Puig and motivated by the Sylow theorems for finite groups. Normal fusion subsystems of $\mathcal{F}$ are defined by analogy with normal subgroups of a group, and $\mathcal{F}$ is simple if it has no nontrivial proper normal subsystems.

One special case which yields many interesting examples is that where $\mathcal{F}$ is simple and $S$ contains a (unique) abelian subgroup $A In this talk, I want to focus on my more recent work with Albert Ruiz, where we described simple fusion systems$\mathcal{F}$over nonabelian$p$-groups that contain an abelian subgroup$A$of index$p$that is not elementary abelian. It turns out that for such$\mathcal{F}$,$A$is homocyclic except in a few cases where$\dim(\Omega_1(A))=p-1$. Also, when$A$is homocyclic,$\mathcal{F}$is determined by$\textup{Aut}_{\mathcal{F}}(A)$, its action on$\Omega_1(A)$, the exponent of$A$, and the$\mathcal{F}$-essential subgroups. We also looked at the limiting’’ case where$S$is infinite, and$A$is a product of copies of$\Z/p^\infty\$.

This talk is part of the Algebra Seminar series.