On blocks of finite groups with a certain factorization (Q1057360)

Let R be a complete discrete valuation ring such that its residue class field is algebraically closed of prime characteristic p. Furthermore, let G be a finite group with factorization \(G=C_ G(W)WO_{p'}(G)\) for some p-subgroup W of G. Then the Brauer correspondence induces a bijection between the set of blocks of \(C_ G(W)W\) and the set of blocks of G with some defect group containing W. We show that corresponding blocks are Morita equivalent thus giving an easy proof of a special case of a result announced by \textit{E. C. Dade} [Proc. Symp. Pure Math. 37, 401-403 (1980; Zbl 0456.20002)].