A splitting theorem for blocks

Abstract

Let F be an algebraically closed field of prime characteristic p, let G be a finite group, and let H be a normal subgroup of G such that G/H is a ^-group. Moreover, let B be a block of the group algebra FH of H over F. By Osima's theorem, there is a unique block A of FG covering B. We are interested in the structure of A. As usual, the general case reduces to the special one where B is (/-stable. Thus we assume in the following that B is G-stable and denote by P a defect group of A. Then Q\—Pr\H is a defect group of B, and G = F//(see [3, V]). If P is abelian then the character theory of A is described in a paper by R. Knόrr [5]. We are interested in the structure of A as a ring under the additional hypothesis that Q has a complement in P. We prove that such a splitting of defect groups implies a splitting of blocks: