Exotic Fusion Systems Research Articles

Proving a conjecture for fusion systems on a class of groups

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≥5. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups. Considering small primes, there are no exotic fusion systems on 2-groups of maximal class and for p=3, we prove block-exoticity of two exotic fusion systems described by Diaz–Ruiz–Viruel.

Punctured groups for exotic fusion systems

AbstractThe transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the ‐local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are, however, not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite ‐group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson–Solomon exotic fusion system at the prime 2 has a punctured group, while the others do not. As for exotic fusion systems at odd primes , we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial ‐groups of order .

Fusion systems with Benson–Solomon components

The Benson–Solomon systems comprise an infinite ascending family of simple exotic fusion systems at the prime 2. The results we prove give significant additional evidence that these are the only simple exotic 2-fusion systems, as conjectured by Solomon. We consider a saturated fusion system F having an involution centralizer with a component C isomorphic to a Benson–Solomon fusion system, and we show under rather general hypotheses that F cannot be simple. Furthermore, we prove that if F is almost simple with these properties, then F is isomorphic to the next larger Benson–Solomon system extended by a group of field automorphisms. Our results are situated within Aschbacher’s program to provide a new proof of a major part of the classification of finite simple groups via fusion systems. One of the most important steps in this program is a proof of Walter’s Theorem for fusion systems, and our first result is specifically tailored for use in the proof of that step. We then apply Walter’s Theorem to treat the general Benson–Solomon component problem under the assumption that each component of an involution centralizer in F is on the list of currently known quasisimple 2-fusion systems.

Fusion systems on a Sylow p-subgroup of SU4(p)

We determine, for p odd, all saturated fusion systems F on a Sylow p-subgroup S of the unitary group SU4(p) and we prove that they are all realizable by finite groups. In particular, we prove that S does not support any exotic fusion systems.

Block-exoticity of a family of exotic fusion systems

Abstract We prove that each exotic fusion system ℱ {\mathcal{F}} on a Sylow p-subgroup of G 2 ⁢ ( p ) {G_{2}(p)} for an odd prime p with 𝒪 p ⁢ ( ℱ ) = 1 {\mathcal{O}_{p}(\mathcal{F})=1} is block-exotic. This gives evidence for the conjecture that each exotic fusion system is block-exotic. We prove two reduction theorems for block-realisable fusion systems.

Fusion Systems over Mp(1, 1, 1) × A-groups

In this paper we study the saturated fusion systems over a direct product of the extraspecial group of order p3 of exponent p and a finite abelian p-group. The result provides some new exotic fusion systems, whose unique components are isomorphic to the exotic fusion systems over $$7_ + ^{1 + 2}$$ found by Ruiz and Viruel.

On exotic saturated fusion systems

In this paper, we prove that a product F1 × F2 of saturated fusion systems is exotic if and only if at least one of the factors is exotic. This result provides a method to construct new exotic fusion systems by known exotic fusion systems. We also investigate central products of saturated fusion systems.

A Family of fusion systems related to the groups Sp4(p a ) and G2(p a )

A family of exotic fusion systems generalizing the group fusion systems on Sylow p-subgroups of G2(pa) and Sp4(pa) is constructed.

Mackey functors and sharpness for fusion systems

We develop the fundamentals of Mackey functors in the setup of fusion systems including an acyclicity condition as well as a parametrization and an explicit description of simple Mackey functors. Using this machinery we extend Dwyer's sharpness results to exotic fusion systems $\mathcal{F}$ on a finite $p$-group $S$ with an abelian subgroup of index $p$.

A spectral sequence for fusion systems

We build a spectral sequence converging to the cohomology of a fusion system with a strongly closed subgroup. This spectral sequence is related to the Lyndon-Hochschild-Serre spectral sequence and coincides with it for the case of an extension of groups. Nevertheless, the new spectral sequence applies to more general situations like finite simple groups with a strongly closed subgroup and exotic fusion systems with a strongly closed subgroup. We prove an analogue of a result of Stallings in the context of fusion preserving homomorphisms and deduce Tate's p-nilpotency criterion as a corollary.

Simple fusion systems over p-groups with abelian subgroup of index p: I

For an odd prime p, we look at simple fusion systems over a finite nonabelian p-group S which has an abelian subgroup A of index p. When S has more than one such subgroup, we reduce this to a case already studied by Ruiz and Viruel. When A is the unique abelian subgroup of index p in S and is not essential (equivalently, is not radical) in the fusion system, we give a complete list of all possibilities which can occur. This includes several families of exotic fusion systems, including some which have proper strongly closed subgroups.

Reduced, tame and exotic fusion systems

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 fusion systems of any finite group. Our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable, and that every reduced fusion system which is not tame is the reduction of some exotic (nonrealizable) fusion system.

Minimal characteristic bisets and finite groups realizing Ruiz–Viruel exotic fusion systems

Continuing our previous work (Park, 2010) [2], we determine a minimal left characteristic biset X for every exotic fusion system F on the extraspecial group S of order 7 3 and exponent 7 discovered by Ruiz and Viruel (2004) [6], and analyze the finite group G obtained from X by the method of Park (2010) [2] which realizes F as the full subcategory of the 7-fusion system of G. In particular, we obtain an upper bound for the exoticity index for F .

Two families of exotic fusion systems

We construct two infinite families of exotic fusion systems.

Saturated fusion systems over $2$-groups

We develop methods for listing, for a given 2-group $S$, all nonconstrained centerfree saturated fusion systems over $S$. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order $2^7$ and two of order $2^{10}$. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order.