simple groups (CFSG)) proofs of numerical properties of cp(G) with explicit (but sometimes crude) bounds, though in one case we use CFSG to sharpen significantly an estimate obtained without it. We show that cp(G) → 0 as either the index or the derived length of the Fitting subgroup of G tends to infinity. Dihedral 2-groups illustrate that the same is not true for the nilpotence class. The third objective is to point out that the solution of the (coprime) k(GV )-problem can lead to some quite strong information about cp (we mostly use this for solvable groups in which case the results do

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Abstract. We study, via character-theoretic methods, an -analogue of the modular representation theory of the symmetric group, for an arbitrary integer ≥ 2. We find that many of the invariants of the usual block theory (ie. when is prime) generalize in a natural fashion to this new context.The study of the modular representation theory of symmetric groups was initiated in the 1940's. One of the first highlights was the proof of the so-called Nakayama conjecture describing the distribution of the irreducible characters into p-blocks in terms of a combinatorial condition on the partitions labelling them. More specifically two irreducible characters are in the same p-block if and only if the partitions labelling them have the same p-core. There is also a comprehensive literature on decomposition numbers, Cartan matrices and other block-theoretic invariants of symmetric groups.The representation theory of symmetric groups has served as a source of inspiration for the study of representations of other classes of groups and algebras. As an example we may refer to the book [9]. Corollary 5.38 in that book presents an analogue of the Nakayama conjecture for Iwahori-Hecke algebras for the symmetric group S n at an -th root of unity. Donkin [4] has presented a direct link between the representation theory of these algebras and an -analogue of the modular representation theory of the symmetric groups. It thus seems a natural problem to study " -blocks" of S n . We attempt to do this here based primarily on the ordinary character theory of symmetric groups and on some very general ideas from the character theory of finite groups. We study analogues of blocks, of the second main theorem on blocks, of decomposition matrices and of Cartan matrices in this context and prove an -analogue of the Nakayama conjecture. We believe that this approach may provide additional insight, eg. concerning the invariant factors of Cartan matrices. For instance we show that these calculations for a given block of weight w may be performed inside the wreath product Z S w . It should be mentioned that Brundan and Kleshchev [3] have recently given a formula for the determinant of the Cartan matrix of an -block for the Hecke algebras. In view of [4] this also is the determinant of the Cartan matrix of an -block of S n . (See Proposition 6.10 for details).The paper is organized as follows: The first two sections present a very general theory of contributions, perfect isometries, sections and blocks, suitable for our purposes. These sections may have independent interest beyond the questions at hand. In section 3 we introduce -sections and -blocks in symmetric groups and prove an analogue of the second main theorem of blocks. Then in section 4 we construct "basic sets", i.e. integral bases for the restrictions of the generalized

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We 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.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 E-mail address: g.r.robinson@abdn.ac.uk. 1 Since 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.