Related papers: Fusion systems with Benson-Solomon components

The Benson-Solomon systems comprise the only known family of simple saturated fusion systems at the prime two that do not arise as the fusion system of any finite group. We determine the automorphism groups and the possible almost simple…

We study a saturated fusion system F on a finite 2-group S having a Baumann component based on a dihedral 2-group. Assuming F is 2-perfect with no nontrivial normal 2-subgroups, and the centralizer of the component is a cyclic 2-group, it…

To each pair consisting of a saturated fusion system over a $p$-group together with a compatible family of K\"ulshammer-Puig cohomology classes, one can count weights in a hypothetical block algebra arising from these data. When the pair…

It is a long-standing open problem raised by Starostin to describe all finite groups with soluble centralizers of involutions. One can observe that if the centralizer fusion system of an involution is nilpotent, then the centralizer of that…

Given a saturated fusion system $\mathcal{F}$ over a $2$-group $S$, we prove that $S$ is abelian provided any element of $S$ is $\mathcal{F}$-conjugate to an element of $Z(S)$. This generalizes a Theorem of Camina--Herzog, leading to a…

Brauer and Fowler noted restrictions on the structure of a finite group G in terms of the order of the centralizer of an involution t in G. We consider variants of these themes. We first note that for an arbitrary finite group G of even…

We prove, when $S$ is a $2$-group of order at most $2^9$, that each reduced fusion system over $S$ is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a $2$-group of order at…

We prove that if $\mathcal{E}\trianglelefteq\mathcal{F}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S(\mathcal{E})\le T$, and either $Aut_{\mathcal{F}}(T)/Aut_{\mathcal{E}}(T)$ or $Out(\mathcal{E})$ is…

When $(S,\mathcal{F},\mathcal{L})$ is a $p$-local finite group and $(T,\mathcal{E},\mathcal{\L}_0)$ is weakly normal in $(S,\mathcal{F},\mathcal{L})$ we show that a definition of $C_S(\mathcal{E})$ given by Aschbacher has a simple…

Aschbacher's program for the classification of simple fusion systems of "odd" type at the prime 2 has two main stages: the classification of 2-fusion systems of subintrinsic component type and the classification of 2-fusion systems of…

We study a certain family of simple fusion systems over finite $3$-groups, ones that involve Todd modules of the Mathieu groups $2M_{12}$, $M_{11}$, and $A_6=O^2(M_{10})$ over $\mathbb{F}_3$, and show that they are all isomorphic to the…

The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups,…

Let $q$ be a power of a fixed prime $p$. We classify up to isomorphism all simple saturated fusion systems on a certain class of $p$-groups constructed from the polynomial representations of $\mathrm{SL}_2(q)$, which includes the Sylow…

We correct an error in Lemma 3.1 of my paper coauthored with Ran Levi on the Benson-Solomon fusion systems, and show that the change does not affect any of the other results in that paper. More precisely, as pointed out to us by Justin…

We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular,…

In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here we verify some of these…

We define here two new classes of saturated fusion systems, reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the…

We study higher limits over the centric orbit category of a fusion system realized by an amalgamated product. In so doing we provide a novel technique for studying the Diaz-Park sharpness conjecture and prove it (in the case of the…

We describe several exotic fusion systems related to the Sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow $3$-subgroups of the Conway group $\mathrm{Co}_1$ and the Thompson group…

We determine for which known finite simple groups $G$ and which primes $p$ the $p$-fusion system of $G$ is simple. This means first collecting together the results that were already known (and correcting two errors made in an earlier study…