Amalgams, blocks, weights, fusion systems and finite simple groups

AbstractWe associate an iterated amalgam of finite groups to a certain class of fusion systems on finite p-groups (p a prime), in such a way that the p-group of the fusion system is a maximal finite p-subgroup of the resulting group, unique up to conjugacy, and, furthermore, the conjugation action of the resulting (usually infinite) group on p-subgroups induces the original fusion system on the p-group.1 In view of earlier work of Puig and of Broto, Castellana, Grodal, Levi and Oliver [C. Broto, N. Castellana, J. Grodal, R. Levi, R. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. (3) 91 (2005) 325–354], the fusion systems we deal with include all saturated fusion systems. 1Since this paper was written, we have been informed that I. Leary and R. Stancu are currently writing a paper with a different construction to realise a fusion system on a finite p-group via a group. Their work and ours are independent of each other.The resulting amalgam has free normal subgroups of finite index, and we examine the images of the group by its maximal free normal subgroups of finite index; these images all contain (isomorphic copies of) the original p-group (and are generated by the (images of the) finite groups used in the amalgamation). If there is no non-trivial normal p-subgroup of the fusion system (equivalently, if the iterated amalgam constructed has no non-trivial normal p-subgroup), then the generalised Fitting subgroup of each of these homomorphic images is a direct product of non-Abelian simple groups, each of order divisible by p (and if there is no proper non-trivial strongly closed p-subgroup in the fusion system, then the generalised Fitting subgroup of any of these finite groups is characteristically simple).We note that the (finite-dimensional) representation theory of this amalgam is (almost by construction) p-locally determined. In the case of the fusion system associated to a p-block of a finite group, we suggest strong links between block-theoretic invariants of the above finite epimorphic images of the associated amalgam and of the original block. We believe that the results of this paper offer the prospect of at least the beginnings of a structural explanation for some of the current numerical conjectures in block theory (e.g. [J.L. Alperin, Weights for finite groups, in: Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, 1987, pp. 369–379; E.C. Dade, Counting characters in blocks, I, Invent. Math. 109 (1) (1992) 187–210])