Saturated fusion systems with parabolic families

Abstract

Let G be group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N G (S).Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system. We give an application to fusion systems with parabolic families of classical type.

Saturated fusion systems on $p$-groups of maximal class

“…This is possibly be an illusion created by considering groups which are is some way small. Evidence that they may be exotic after all comes, for example, from [46] where it is shown that, in certain good situations, a saturated fusion system determines a locally finite classical Tits chamber system and so is not exotic. Work of van Beek generalizing the classification of groups with a weak BN-pair has also not revealed any surprises [55].…”

Exotic Fusion Systems Related to Sporadic Simple Groups

“…The following proposition is proved using a result of Onofrei [Ono11] which identifies a parabolic system in F, and further restrictions then identify Sp 6 (3) from an associated chamber system. We will not define parabolic systems for fusion systems here, but we remark that in the case of parabolic systems in groups, the definition is meant to abstractly capture a set of minimal parabolics containing a "Borel", in analogy with groups of Lie type in defining characteristic.…”

“…In the case of the group G := Co 1 , the groups N G (E i ) for i ∈ {1, 2, 3} all contain the "Borel" N G (S) and together generate G and so successfully form something akin to a parabolic system. Utilized above, work by Onofrei [Ono11] parallels the group phenomena in fusion systems and provides conditions in which a parabolic system within a fusion system F gives rise to a parabolic system in the group sense. The resulting completion of the group parabolic system realizes the fusion system and if certain additional conditions are satisfied, the fusion system is saturated.…”