Related papers: Exotic Fusion System on a Subgroup of the Monster

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional $p$-groups of maximal nilpotency class, where $p \geq 5$. This is done by considering a family of exotic fusion systems…

We prove that each exotic fusion system $\mathcal F$ on a Sylow $p$-subgroup of $G_2(p)$ for an odd prime $p$ with $\mathcal O_p(\mathcal F)=1$ is block-exotic. This gives evidence for the conjecture that each exotic fusion system is…

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 generalize the Reduction Theorem of Kessar-Stancu so it can be applicable to exotic fusion systems over the maximal nilpotency class of rank two $3$-groups. This is an essential step towards proving that these fusion systems are also…

We study saturated fusion systems on $p$-groups having sectional rank $3$ for all odd primes $p$. For $p\geq 5$, we obtain a complete classification of the ones that do not have any non-trivial normal $p$-subgroups.

Let $p$ be an odd prime and $S$ a nonabelian finite $p$-group. In [9, 10], they proposed the following conjecture: if $\mathcal{F}$ be a transitive fusion system over a finite $p$-group $S$, then $S$ is either extraspecial of order $p^{3}$…

We classify all corefree fusion systems on a Sylow $3$-subgroup of the sporadic groups $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$ and $\mathrm{B}$. We show that the $3$-group in each case does not support any exotic fusion systems. This is the…

Suppose that all nontrivial subsections of a $p$-block $B$ are conjugate (where $p$ is a prime). By using the classification of the finite simple groups, we prove that the defect groups of $B$ are either extraspecial of order $p^3$ with $p…

Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on…

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…

For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems…

An $\mathcal{F}$-essential subgroup is called a pearl if it is either elementary abelian of order $p^2$ or non-abelian of order $p^3$. In this paper we start the investigation of fusion systems containing pearls: we determine a bound for…

Continuing our previous work, we determine a minimal left characteristic biset $X$ for every exotic fusion system $\mathcal{F}$ on the extraspecial group $S$ of order $7^3$ and exponent $7$ discovered by Ruiz and Viruel, and analyze the…

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety…

A family of exotic fusion systems generalizing the group fusion systems on Sylow $p$-subgroups of $\mathrm{G}_2(p^a)$ and $\mathrm{Sp}_4(p^a)$ is constructed.

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…

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,…

We determine, for $p$ odd, all saturated fusion systems on a Sylow $p$-subgroup $S$ of the unitary group $SU_4(p)$ and we prove that they are all realizable by finite groups. In particular, we prove that $S$ does not support any exotic…

As a contribution to an eventual solution of the problem of determination of the maximal subgroups of the Monster we show that there is a unique conjugacy class of subgroups isomorphic to $PSU_3(8)$. The argument depends on some…

We determine the conjugacy class fusion from certain maximal subgroups of the Monster to the Monster, to justify the addition of these data to the Character Table Library in the computational algebra system GAP. The maximal subgroups in…