Related papers: Fusion systems on small p-groups

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.

We complete the determination of saturated fusion systems on maximal class 3-groups of rank two.

We consider saturated fusion systems $\mathcal F$ on a Sylow $2$-subgroup of $\Omega^+_8(2)$ with $O_2(\mathcal F) = 1$. Examples for this are the $2$-fusion systems of $\Omega^+_8(2)$, $\Omega^+_8(2):3$, $P\Omega^+_8(3)$ and…

We determine all reduced saturated fusion systems supported on a finite $p$-group of nilpotency class two. As a consequence, we obtain a new proof of Gilman & Gorenstein's classification of finite simple groups with class two Sylow…

For a prime $p$, we describe a protocol for handling a specific type of fusion system on a $p$-group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or…

We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups (see [Craven-Glesser, 2012] and [Sambale,…

For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of $\mathrm{G}_2(p^n)$ or $\mathrm{PSU}_4(p^n)$ with $n\in\mathbb{N}$, we determine all saturated fusion systems supported on $S$ up to isomorphism.

For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of a rank $2$ simple group of Lie type in characteristic $p$, we determine all saturated fusion systems supported on $S$ up to isomorphism.

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 give another proof of an observation of Th\'evenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system $\CF$ on a finite $p$-group $S$, we show that the number of the $\CF$-conjugacy classes of…

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…

In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analisis of the possible saturated fusion systems in terms of the outer automorphism group ant the proper F-radical subgroups.…

Building upon previous results, a classification is given of finite $p$-groups of which subgroups of order $p$ are all fused. This completes the classification problem dated back to Higman 1963 on the so-called Suzuki $2$-groups, and…

We show that every (not necessarily saturated) fusion system can be realized as a full subcategory of the fusion system of a finite group. This result extends our previous work \cite{Park2010} and complements the related result…

We define minimal fusion systems in a way that every non-solvable fusion system has a section which is minimal. Minimal fusion systems can also be seen as analogs of Thompson's N-groups. In this paper, we consider a minimal fusion system…

For a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if it has nilpotency class $n-1$. Here we examine saturated fusion systems on maximal class $p$-groups and, in particular, we describe all the reduFor a prime…

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains…

We classify fusion systems $\mathcal{F}$ in which $O_p(\mathcal{F})=\{1\}$, and there are two $\mathrm{Aut}_{\mathcal{F}}(S)$-invariant essential subgroups whose normalizer systems generate $\mathcal{F}$. We employ the amalgam method and,…

Saturated fusion systems are categories modeling properties of conjugacy of p-subgroups in finite groups. It was shown by Chermak that they correspond nicely to group-like structures called localities. In this paper we start to explore how…

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…