On TI and TI Defect Blocks

We show that every trivial intersection block of a finite group (as introduced by J. L. Alperin and M. Broué (1979, Ann. of Math.110, 143–157)) has trivial intersection (TI) defect groups but that the converse is not true in general. We then present some conditions equivalent to B being a TI block, generalizing the idea of a k-generated p-core to B-subgroups. In particular we give further weight to Olssonʹs assertion that TI blocks are a better generalization of groups with TI Sylow p-subgroups than are TI defect blocks. Finally we describe the rôle of the generalized k-generated p-core in the control of fusion of subpairs.