EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL

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. Let G be a nite group, with Sylow p-subgroup S 2 Sylp(G). The fusion system of G (at p) is the category FS(G) whose objects are the subgroups of G, and where MorFS(G)(P;Q) is the set of monomorphisms from P to Q induced by conjugation by elements of G. The transporter system of G at p is the category TS(G) with the same objects as FS(G), and with morphism sets MorTS(G)(P;Q) = NG(P;Q): the set of elements of G which conjugate P into Q. A subgroup P S is called p-centric in G if CG(P ) = Z(P ) C 0 G(P ) for a (unique) subgroup C 0 G(P ) of order prime to p; and the centric linking system of G is the category LS(G) whose objects are the subgroups of S which are p-centric in G, and where MorLS(G)(P;Q) = NG(P;Q)=C 0 G(P ). All of these de nitions are repeated in more detail at the beginning of Section 1. In several papers, such as [BLO1] and [O2], the fusion and linking systems of G are shown to play a central role in describing homotopy theoretic properties of the p-completed classifying space BGp . Abstract fusion and linking systems have also been de ned and studied, and are shown in [BLO2] to have many of the same properties as the fusion and linking systems of nite groups. A p-local nite group is de ned to be a triple (S;F ;L), where S is a nite p-group, F is a saturated fusion system over S (De nitions 1.2 and 1.3), and L is a centric linking system associated to F (De nition 1.6). Normal and central p-subgroups of fusion systems and linking systems are also de ned (De nition 1.4). Certain types of extensions of p-local nite groups, and in particular central extensions, were studied in [BCGLO2]. One hope was that extensions could provide a new way to construct exotic examples. But in the case of central extensions, this was shown to be impossible. By [BCGLO2, Theorem 6.13 and Corollary 6.14], if A is a central subgroup in (S;F ;L), and (S=A;F=A;L=A) is induced by a group G, then (S;F ;L) is induced by a group e G such that A Z( e G) and e G=A = G. In this paper, we look at the more general situation of extensions with p-group kernel. Equivalently, given a p-local nite group (S;F ;L) and a nite p-group A, we want to nd p-local nite groups (e S; e F ; e L) such that A C F and (S;F ;L) = (e S=A; e F=A; e L=A). One problem when doing this is that the fusion system F=A contains too little information: F cannot be described as an extension of F=A by A in any sense. 2000 Mathematics Subject Classi cation. Primary 55R35. Secondary 55R40, 20D20.