On blocks of finite groups with radical cube zero (Q1086344)

Let G be a finite group and let K be an algebraically closed field of prime characteristic p. Let B be a block of the group algebra KG with defect group D and Jacobson radical J(B). It is proved that \([J(B)]^ 3=0\), \([(J(B)]^ 2\neq 0\) if and only if one of the following holds; (1) \(p=2\), D is a four-group and B is isomorphic to a matrix ring over KD or B is a Morita equivalent to \(KA_ 4\), where \(A_ 4\) is the alternating group of degree 4, or (2) \(p\neq 2\), \(| D| =p\), the number of simple KG-modules in B is p-1 or (p-1)/2 and the Brauer tree of B is a straight line segment such that the exceptional vertex is an end point (if it exists).