On the source algebra equivalence class of blocks with cyclic defect groups, III

Abstract

This series of papers is a contribution to the program of classifying $p$-blocks of finite groups up to source algebra equivalence, starting with the case of cyclic blocks. To any $p$-block $\mathbf{B}$ of a finite group with cyclic defect group $D$, Linckelmann associated an invariant $W( \mathbf{B} )$, which is an indecomposable endo-permutation module over $D$, and which, together with the Brauer tree of~$\mathbf{B} $, essentially determines its source algebra equivalence class. In Part II of our series, assuming that $p$ is an odd prime, we reduced the classification of the invariants $W( \mathbf{B} )$ arising from cyclic $p$-blocks $\mathbf{B}$ of quasisimple classical groups to the classification for cyclic $p$-blocks of quasisimple quotients of special linear or unitary groups. This objective is achieved in the present Part III.