[For the entire collection see Zbl 0699.00023.]
From the author’s abstract. “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”. Theorem. Let S be an unramified (and integrally closed) finite extension of \({\mathbb{Z}}_ p\). Let B be the principal block of group algebras SG, SH over S, for finite groups G and H. Let D and E be Sylow p-subgroups of G and H, respectively, and identify them with their projections onto B. Assume that E is normalized in the sense of mapping to 1 under the augmentation \(B\to S\) induced by SG\(\to S\) (which certainly also sends D to 1). Assume also that D is cyclic (which implies that E is cyclic), and that D and E are T.I. sets in G and H, respectively. Then D is conjugate to E in B. In a late note insertion to the paper, the author states that the T.I. set assumption in the Theorem is not necessary and that the details will appear elsewhere.