On two theorems of Flavell

Abstract

We extend two theorems due to P. Flavell [6] to arbitrary fusion systems. The fusion system of a finite group G at a prime p encodes the structure of a Sylow-p-subgroup S together with extra information on G-conjugacy within S in category theoretic terms. From work of Alperin and Broue [1] it emerged that fusion systems of finite groups are particular cases of fusion systems of p-blocks of finite groups. In the early 1990’s, Puig introduced the notion of an abstract fusion system on a finite p-group S as a category whose objects are the subgroups of S and whose morphism sets satisfy a list of properties modelled around what one observes in the case of fusion systems of finite groups and blocks. There are ‘exotic’ fusion systems which do not arise as fusion system of a block. Benson [2] suggested that nonetheless any fusion system should give rise to a p-complete topological space which should coincide with the p-completion of the classifying space BG in case the fusion system does arise as fusion system of a finite group G. Broto, Levi and Oliver laid in [4] the homotopy theoretic foundations of such spaces called p-local finite groups and gave in particular a cohomological criterion for the existence and uniqueness of a p-local finite group associated with a given fusion system. While the existence and uniqueness of p-local finite groups for arbitrary fusion systems is still an open problem, there has been in recent years a steadily growing body of work by many authors trying to add to the understanding of fusion systems by extending classical concepts and results on the p-local structure of finite groups (some of which were relevant for the classification of finite simple groups) to all fusion systems. This is also the underlying philosophy of the present note. The following two theorems are Flavell’s Theorem A and Theorem B in [6], extended to arbitrary fusion systems. Our general terminology on fusion systems follows [7]; in particular, by a fusion system we always mean a saturated fusion system. Theorem 1. Let p be an odd prime, S a finite p-group, F a fusion system on S and α an automorphism of S acting freely on S−{1}, which stabilises F and whose order is a prime number r which does not divide the orders of the automorphism groups AutF(R) for all F-centric radical subgroups R of S. Then F = NF(S). The hypothesis that α stablises F means that for any two subgroups Q, R of S and any morphism φ : Q → R in F the morphism α ◦ φ ◦ α|α(Q) : α(Q) → α(R) is again a morphism in F . Moreover, a subgroup Q of S is called F -centric if CS(Q ) = Z(Q) for all subgroups Q of S such that Q ∼= Q in F ; a subgroup Q of S is called F -radical if Op(AutF (Q)) = AutQ(Q), the group of inner automorphisms of Q. By Alperin’s fusion theorem, F is completely determined by the automorphism groups in F of F -centric radical subgroups of S.