Abstract. 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.
Abstract. We study extensions of p-local nite groups where the kernel is a pgroup. In particular, we construct examples of saturated fusion systems F which do not come from nite groups, but which have normal p-subgroups A C F such that F=A is the fusion system of a nite group. One of the tools used to do this is the concept of a \transporter system", which is modelled on the transporter category of a nite group, and is more general than a linking system.vet G e nite groupD with ylow pEsugroup S P yl p @GAF he fusion system of G @t pA is the tegory p S @GA whose ojets re the sugroups of GD nd where wor p S @GA @P; QA is the set of monomorphisms from P to Q indued y onjugtion y elements of GF he transporter system of G t p is the tegory S @GA with the sme ojets s p S @GAD nd with morphism sets wor S @GA @P; QA a N G @P; QAX the set of elements of G whih onjugte P into QF e sugroup P S is lled pEentri in G ifG @P A of order prime to pY nd the centric linking system of G is the tegory v c S @GA whose ojets re the sugroups of S whih re pEentri in GD nd where wor v c S @GA @P; QA a N G @P; QA=C H G @P AF ell of these denitions re repeted in more detil t the eginning of etion IF sn severl ppersD suh s fvyI nd yPD the fusion nd linking systems of G re shown to ply entrl role in desriing homotopy theoreti properties of the pEompleted lssifying spe BG p F estrt fusion nd linking systems hve lso een dened nd studiedD nd re shown in fvyP to hve mny of the sme properties s the fusion nd linking systems of nite groupsF e p-local nite group is dened to e triple @S; p; vAD where S is nite pEgroupD p is saturated fusion system over S @henitions IFP nd IFQAD nd v is centric linking system ssoited to p @henition IFTAF xorml nd entrl pEsugroups of fusion systems nd linking systems re lso dened @henition IFRAF gertin types of extensions of pElol nite groupsD nd in prtiulr entrl extenE sionsD were studied in fgqvyPF yne hope ws tht extensions ould provide new wy to onstrut exoti exmplesF fut in the se of entrl extensionsD this ws shown to e impossileF fy fgqvyPD heorem TFIQ nd gorollry TFIRD p=A; e v=AAF yne prolem when doing this is tht the fusion system p=A ontins too little informtionX p nnot e desried s n extension of p=A y A in ny senseF
Abstract. 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.
Abstract. We study reduced fusion systems from the point of view of their essential subgroups, using the classification by Goldschmidt and Fan of amalgams of prime index to analyze certain pairs of such subgroups. Our results are applied here to study reduced fusion systems over 2-groups of order at most 64, and also reduced fusion systems over 2-groups having abelian subgroups of index two. More applications are given in later papers.A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are monomorphisms between subgroups, and which satisfy certain axioms first formulated by Puig [Pg2] and motivated by conjugacy relations among p-subgroups of a given finite group. A saturated fusion system is reduced if it has no proper normal subsystem of p-power index, no proper normal subsystem of index prime to p, and no nontrivial normal p-subgroup. (All three of these concepts are defined by analogy with finite groups.) Reduced fusion systems need not be simple, in that they can have proper nontrivial normal subsystems. They were introduced by us in [AOV] as forming a class of fusion systems which is small enough to be manageable, but still large enough to detect any fusion systems (reduced or not) which are "exotic" (not defined via conjugacy relations in any finite group).When G is a finite group and S ∈ Syl p (G), the version of Alperin's fusion theorem shown by Goldschmidt [Gd1] says that all G-conjugacy relations among subgroups of S are generated by Aut G (S) (automorphisms induced by conjugation in G), together with Aut G (P ) for certain "essential" proper subgroups of S, and restrictions of such automorphisms. There is a version of this result for abstract fusion systems (see Theorem 1.2), which says that a fusion system F is generated by F-automorphisms of F-essential subgroups (Definition 1.1). Our goal in this and our other papers is to study, and to classify in certain cases, reduced fusion systems from the point of view of their essential subgroups and generating automorphisms.This point of view was introduced in [OV], where two of us described how fusion systems over a given 2-group S could be classified by first listing the subgroups of S which potentially could be essential, using Bender's theorem on groups with strongly embedded subgroups. When we try to extend those methods to larger classes of groups, it is useful to search for pairs of essential subgroups via theorems of Goldschmidt and Fan classifying certain types of amalgams.
In this paper we compute some derived functors Ext of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.When the group is a cyclic 2-group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor Ext.When the group is G = Z/2×Z/2, we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of G by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired Ext functors.
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