The nilpotency index of the radicals of blocks with abelian defect groups

Abstract

LetF be an algebraically closed field of characteristicp > 0, letG be a finitep-solvable group and letB be a block of the group algebraFG with abelian defect groupD. In this note we provide a precise formula for the nilpotency indext(B) ofJ(B), whereJ(B) is the Jacobson radical ofB. Namely, we prove that if $$D \cong Z_{p^{n_1 } } \times Z_{p^{n_2 } } \times \ldots \times Z_{p^{n_k } }$$ then $$t(B) = 1 - k + \sum\limits_{i = 1}^k {p^{n_i } .}$$