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Morita equivalent is well known ; this follows in particular from Dade's work in [5] or from Puig's structure theorem in [14] on blocks of finite p-solvable groups that describes the source algebras of such blocks, independently of whether P is abelian or not, and also implies Corollary 1.2. Moreover, Broue! 's conjecture [1, 3.7] (asserting that, for any finite group G having an abelian Sylow-p-subgroup P, the principal blocks of G and N G (P) are splendidly derived equivalent) holds if G is p-solvable, since in that case the corresponding block algebras are actually naturally isomorphic (cf. [1, 1.4, 3.7]). Our theorem asserts that we have splendid derived equivalences for corresponding non-principal blocks.In fact, Theorem 1.1 will be a formal consequence of some of Puig's methods and of Rickard's idea of describing endo-permutation modules as the degree 0 homology of certain bounded complexes of permutation modules called endo-split p-permutation resolutions. In this paper, we restrict ourselves to blocks with abelian defect groups, since at this stage it is not known in general whether all endopermutation kP-modules for possibly non-abelian P arising from a block of a finite p-solvable group have endo-split p-permutation resolutions.Also, Rickard's technique is not limited to finite p-solvable groups, as can be easily seen from the proof we give below, since the main idea is to show the following. Given a finite p-group P, an endo-permutation P-module V having an endo-split p-permutation resolution and A an interior P-algebra satisfying suitable conditions, and then setting S l End (V ), the natural Morita equivalence between A and

In this paper, we obtain information about the minimal degree δ of any non-trivial projective representation of the group PSL(n, q) with n ≧ 2 over an arbitrary given field K. Our main results for the groups PSL(n, q) (Theorems 4.2, 4.3, and 4.4) state that, apart from certain exceptional cases with small n, we have the following rather surprising situation: if q = pf (where p is a prime integer) and char K = p, then δ = n, but if q = pf and char K ≠ p, then δ is of a considerably higher order of magnitude, namely, δ is at least qn–l – 1 or if n = 2 and q is odd. Note that for n = 2, this lower bound for δ is the best possible. However, for n ≧ 3, this lower bound can conceivably be improved.

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