Defect Groups and the Isomorphism Problem

As a ring, a block may arise from more than one finite group. The resulting conjugacy issue for defect groups is important for the group ring isomorphism problem and for understanding block theory in general. There are even indirect structural consequences for finite groups, through G. Robinson’s work on the “Z-star theorem” for odd primes. A positive answer to the defect group conjugacy problem is given here for the principal block in the case of cyclic, T.I., Sylow p-subgroups. In a talk at Arcata [S] I raised the following question regarding defect groups: Let B be a block of group rings IpG and IpH, where HP denotes the ring of p-adic integers and G, H are (possibly nonisomorphic) finite groups. Suppose B has defect group D in G and E in H. Identify D and E with their projections on B. Is it then true that, after applying some suitable normalization process to D and E (preserving their isomorphism types), these groups must then be conjugate by a unit of B? In case B is a principal block, normalization should just be the familiar and innocuous projection onto the units of augmentation 1, using an augmentation B + S obtained, say, from the sum-of-coefficients map SG + S. For other blocks the correct formulation of normalization remains part of the problem. iThis research was supported in part by NSF, especially NSF-INT. S. M. F. AstPriSqUe 181-182 (1990)