The nilpotency index of the radicals of blocks with abelian defect groups (Q1093729)

Let F be an algebraically closed field of characteristic \(p>0\), let G be a finite p-solvable group and let B be a block of the group algebra FG with abelian defect group D. In this note we provide a precise formula for the nilpotency index t(B) of J(B), where J(B) is the Jacobson radical of B. Namely, we prove that if \(D\cong Z_{p^{n_ 1}}\times Z_{p^{n_ 2}}\times...\times Z_{p^{n_ k}}\) then \(t(B)=1-k+\sum ^{k}_{i=1}p^{n_ i}\).