University of Birmingham > Talks@bham > Algebra Seminar > Simple fusion systems over p-groups with unique abelian subgroup of index p

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

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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.

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