On the definition of saturated fusion systems (original) (raw)

On the focal subgroup of a saturated fusion system

Journal of Algebra, 2016

The influence of the cyclic subgroups of order p or 4 of the focal subgroup of a saturated fusion system F over a p-group S is investigated in this paper. Some criteria for normality of S in F as well as necessary and sufficient conditions for nilpotency of F are given. The resistance of a p-group in which every cyclic subgroup of order p or 4 is normal, and earlier results about p-nilpotence of finite groups and nilpotency of saturated fusion systems are consequences of our study.

An extension of Glauberman's ZJZJZJ-theorem to fusion systems

2021

Let p be a prime and F be a saturated fusion system over a finite p-group P . The fusion system F is said to be nilpotent if F = FP (P ). We provide new criteria for a saturated fusion system F to be nilpotent, which may be viewed as extending Glauberman’s ZJ-theorem to fusion systems.

Glauberman’s and Thompson’s theorems for fusion systems

Proceedings of The American Mathematical Society, 2008

We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system F on a finite p-group S, and in the cases where p is odd or F is S 4 -free, we show that Z(N F (J(S))) = Z(F) (Glauberman), and that if C F (Z(S)) = N F (J(S)) = F S (S), then F = F S (S) (Thompson). As a corollary, we obtain a stronger form of Frobenius' theorem for fusion systems, applicable under the above assumptions, and generalizing another result of Thompson.

Realisability of p-stable fusion systems

Journal of Algebra, 2019

The aim of this paper is to investigate p-stable fusion systems, where p is an odd prime. We examine realisable fusion systems and prove a generalisation of a result of G. Glauberman. Then we prove that p-stability is determined by the normaliser systems of centric radical subgroups. Finally, we prove that all p-stable fusion systems are realisable provided there exists a stable p-functor.

On p-stability in groups and fusion systems

arXiv (Cornell University), 2017

The aim of this paper is to generalise the notion of p-stability (p is an odd prime) in finite group theory to fusion systems. We first compare the different definitions of p-stability for groups and examine properties of p-stability concerning subgroups and factor groups. Motivated by Glauberman's theorem, we study the question how Qd(p) is involved in finite simple groups. We show that with a single exception a simple group involving Qd(p) has a subgroup isomorphic to either Qd(p) or a central extension of Qd(p) by a cyclic group of order p. Then we define p-stability for fusion systems and characterise some of its properties. We prove a fusion theoretic version of Thompson's maximal subgroup theorem. We introduce the notion of section p-stability both for groups and fusion systems and prove a version of Glauberman's theorem to fusion systems. We also examine relationship between solubility and p-stability for fusion systems and determine the simple groups whose fusion systems are Qd(p)-free.

Existence and uniqueness of classifying spaces for fusion systems over discretep-toral groups

Journal of the London Mathematical Society, 2014

A major questions in the theory of p-local finite groups was whether any saturated fusion system over a finite p-group admits an associated centric linking system, and when it does, whether it is unique. Both questions were answered in the affirmative by A. Chermak, using the theory of partial groups and localities he developed. Using Chermak's ideas combined with the techniques of obstruction theory, Bob Oliver gave a different proof of Chermak's theorem. In this paper we generalise Oliver's proof to the context of fusion systems over discrete p-toral groups, thus positively resolving the analogous questions in p-local compact group theory. * C F where F : C → Ab is a functor from a small category C. The main result of this paper is the following generalisation of [O, Theorem 3.4] to saturated fusion systems over discrete p-toral groups. Theorem A. Let F be a saturated fusion system over a discrete p-toral group S. Then H i (O(F c), Z) = 0 for all i > 0 if p is odd, and for all i > 1 if p = 2. The following result (cf. [Ch], and [O, Theorem A]) now follows from Proposition 1.7 below. Theorem B. Let F be a saturated fusion system over a discrete p-toral group. Then there exists a centric linking system associated to F which is unique up to isomorphism.

On the Basis of the Burnside Ring of a Fusion System

2014

We consider the Burnside ring A(F) of F-stable S-sets for a saturated fusion system F defined on a p-group S. It is shown by S. P. Reeh that the monoid of F-stable sets is a free commutative monoid with canonical basis {α_P}. We give an explicit formula that describes α_P as an S-set. In the formula we use a combinatorial concept called broken chains which we introduce to understand inverses of modified Möbius functions.

The Burnside ring of fusion systems

Advances in Mathematics, 2009

Given a saturated fusion system F on a finite p-group S we define a ring A(F) modeled on the Burnside ring A(G) of finite groups. We show that these rings have several properties in common. When F is the fusion system of G we describe the relationship between these rings.

On fusion rules and solvability of a fusion category

Journal of Group Theory, 2017

We address the question whether or not the condition on a fusion category being solvable is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the