MORITA EQUIVALENT BLOCKS IN NON-NORMAL SUBGROUPS AND p-RADICAL BLOCKS IN FINITE GROUPS

Abstract

Let [Oscr ] be a complete discrete valuation ring with unique maximal ideal J([Oscr ]), let K be its quotient field of characteristic 0, and let k be its residue field [Oscr ]/J([Oscr ]) 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 [mid ]G[mid ]-th roots of unity, where [mid ]G[mid ] 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∈{[Oscr ], k}. 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. Külshammer. However, Külshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Külshammer. Moreover, in the case H is normal in G, we get the same results as Külshammer; however, he uses the results of E. C. Dade, and we do not.