The principal block idempotent
Let F be a field of prime characteristic p, let G be a finite group, let \(B_ 0(FG)\) denote the principal block of FG, and let \(e_ 0(FG)\) denote the corresponding block idempotent. We write \(e_ 0(FG)=\sum_{g\in G}\epsilon_ gg\) with \(\epsilon_ g\in F\) for \(g\in G\). It is known that \(\epsilon_ g=0\) if g is p-singular. We show that \(| G_{p'}| \epsilon_ g=| \{(u,s)\in G_ p\times G_{p'}:\) \(us=g\}| l_ F\) if g is p-regular; here \(G_ p\) and \(G_{p'}\) denote the sets of p-elements and \(p'\)-elements in G. Brauer's Third Main Theorem is an immediate consequence of this result.
scientific article; zbMATH DE number 4130640