ALet be a complete discrete valuation ring with unique maximal ideal J( ), let K be its quotient field of characteristic 0, and let k be its residue field \J( ) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the QGQ-th roots of unity, where QGQ is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R ? o, kq. We consider the following question : when are A and B Morita equivalent ? Actually, we deal with ' naturally Morita equivalent blocks A and B ', which means that A is isomorphic to a full matrix algebra of B, as studied by B. Ku$ lshammer. However, Ku$ lshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Ku$ lshammer. Moreover, in the case H is normal in G, we get the same results as Ku$ lshammer ; however, he uses the results of E. C. Dade, and we do not. Introduction and notationLet (K, , k) be a p-modular system (see [16, p. 230]), that is, is a complete discrete valuation ring with unique maximal ideal J( ) such that J( ) l (π) for π ? , K is the quotient field of of characteristic 0, and k is the residue field \(π) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the QGQ-th roots of unity, where QGQ is the order of G. Then, in particular, K and k are both splitting fields for all subgroups of G (see [16, p. 230 ; 20, p. 13]). Let R ? ok, q. Now let H be an arbitrary subgroup of G, and let A and B respectively be blocks ( p-blocks) of RG and RH (here blocks of RG mean two-sided block ideals of the group algebra RG). We consider the following question, as in Ku$ lshammer [14] : when are A and B Morita equivalent ?As a matter of fact, we investigate naturally Morita equi alent blocks A and B as Ku$ lshammer did in [14]. Here, however, we assume only that H is just a subgroup of G, while he assumed in [14] that H is a normal subgroup of G.In §1 we consider p-radical blocks of finite groups where a result of Hida [9] is generalised (see [6, VI, §6; 21]). In § §2-4 we consider naturally Morita equivalent blocks in non-normal subgroups. In §3 we assume that H is a normal subgroup of G, and we give several results that are generalisations of Ku$ lshammer [14]. We also give a characterisation of an important subgroup G [B] of G which is due to Dade [4], though we do not use Dade's results [4]. In §4 we restrict ourselves to the case that A and B are naturally Morita equivalent of degree one, which is just the case that A and B are isomorphic blocks. Here also we do not assume that H is normal in G. We give an alternative proof of a theorem of Schmid [19].Throughout this paper, we use the following notation and terminology. Let k, , K and π be as above, and let R ? ok, q. Let A be an R-algebra. By an R-algebra A,

Let E be the extraspecial p-group of order p 3 and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence. It is well known that indecomposable summands in the complete stable splitting correspond to simple modules for the mod p double Burnside algebra. We shall use representation theory of the double Burnside algebra and the theory of biset functors. X ij 1991 Mathematics Subject Classification. Primary 55P35, 57T25, 20C20; Secondary 55R35, 57T05.Let Λ be a finite dimensional algebra over a filed k. We denote the Jacobson radical of Λ by rad(Λ), namely, rad(Λ) is the intersection of all maximal ideals of Λ. If e is a primitive idempotent in Λ, then eΛ is a projective indecomposable right Λ-module and eΛ/erad(Λ) is a simple right Λ-module. Let 1 = 1≤i≤l 1≤j≤m(i) e ij be a decomposition of unity into primitive orthogonal idempotents in Λ, where e ij Λ/e ij rad(Λ) ≃ e mn Λ/e mn rad(Λ) if and only of i = m. Thengives the complete set of representatives of isomorphism classes of simple right Λ-modules. Let e i = 1≤j≤m(i) e ij and, in this paper, we call e i an idempotent corresponding to the simple module S i . Multiplication by e i induces the identity map on S i and S j e i = 0 for j = i. On the other hand, we have S i e i1 ≃ Hom Λ (e i1 Λ, S i ) ≃ End Λ (S i ) and the multiplicity m(i) is equal to dim End Λ (Si) S i .Let M be a finite dimensional right Λ-module. Then M has a composition seriesIf every composition factor of M is isomorphic to S i , then M = M e i and the right multiplication by e i induces the identity on M . In general case, to determine M e i , we shall use the following lemma in section 10. Note that H * (E) is not finite dimensional, but each homogeneous part H n (E) is a finite dimensional A p (E, E)module and we can apply the lemma. Lemma 2.1. Let M be a finite dimensional Λ-module and S a simple Λ-module. Let e be an idempotent corresponding to S. Let 0 ⊂ L ⊂ N ⊂ M

Let G be a finite group and H a subgroup. We give an algebraic proof of Mislin's theorem which states that the restriction map from G to H on mod-p cohomology is an isomorphism if and only if H controls p-fusion in G. We follow the approach of P. Symonds [P. Symonds, Mackey functors and control of fusion, Bull. London Math. Soc. 36 (2004) 623-632] and consider the cohomology of trivial source modules.

Abstract.We give a sufficient condition on a p-bloek of a finite group under which the block is /»-radical.