Related papers: Realising fusion systems

For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic,…

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…

We show that every rank two $p$-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group $G$ on a manifold $M$, we construct a smooth free action on…

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…

We prove that the D\'iaz-Park's sharpness conjecture holds for saturated fusion systems defined on a Sylow $p$-subgroup of the group ${\rm G}_2(p)$, for $p\geq 5$.

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…

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…

Linking systems were introduced to provide algebraic models for $p$-completed classifying spaces of fusion systems. Every linking system over a saturated fusion system $\mathcal{F}$ corresponds to a group-like structure called a locality.…

For a saturated fusion system $\mathcal F$ on a $p$-group $S$, we study the Burnside ring of the fusion system $B(\mathcal F)$, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring $B(S)$. We give criteria for…

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…

Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that…

Let $G$ be a finite $p$-group such that $x\Z(G) \subseteq x^G$ for all $x \in G- \Z(G)$, where $x^G$ denotes the conjugacy class of $x$ in $G$. Then $|G|$ divides $|\Aut(G)|$, where $\Aut(G)$ is the group of all automorphisms of $G$.

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 this article, we consider the control of fusion in fusion systems, proving three previously known, non-trivial results in a new, largely elementary way. We then reprove a result of Aschbacher, that the product of two strongly closed…

It is proved that generalized free product of two finite p-groups is a conjugacy p-separable group if and only if it is residually finite p-groups. This result is then applied to establish some sufficient conditions for conjugacy…

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 show that every space that is the union of a `small' family consisting of special P-sets that are F-spaces, is an F-space. We also comment on the sharpness of our results.

It is shown that any finite group $A$ is realizable as the automizer in a finite perfect group $G$ of an abelian subgroup whose conjugates generate $G$. The construction uses techniques from fusion systems on arbitrary finite groups, most…

Suppose $p$ is a prime and $S$ is a Sylow $p$-subgroup of a finite group $G$. If $S$ is normal in $G$, then $Z(S)$ is the direct product of $S \cap Z(G)$ with $[Z(S), G]$. We prove an analogous result for all groups except in some cases…

In this paper, we find some exotic fusion systems which have non-trivial strongly closed subgroups, and we prove these fusion systems are also not realizable by p-blocks of finite groups.