A proof of the Brauer's second main theorem and related results (Q1059711)

The authors have reorganized parts of the standard material in the theory of modular representations of finite groups to obtain a unified proof of Brauer's second main theorem and the following result of Green strengthening Nagao's theorem: Let (K,R,k) be a splitting p-modular system for the subgroups of the finite group G, M an indecomposable RG- lattice belonging to the block B of RG and V an indecomposable component of \(M_ H\) with vertex P belonging to the block b of RH. If \(C_ G(P)\subseteq H\) then \(b^ G=B\). A key ingredient in the proof is the following technical lemma due to \textit{A. Watanabe} [J. Algebra 78, 282- 291 (1982; Zbl 0501.20006)]: Let H be a subgroup of G and let B (resp. b) be a block of RG (resp. RH) with block idempotent E (resp. e). If \(b^ G=B\) then \(eE=e+(1-E)w\) where w is an H-stable linear combination of elements of G not contained in H.