One of the deepest results in modular representation theory is the description of the blocks with a cyclic defect group. This theory was begun by R. Brauer in the early forties with the study of groups with a Sylow p-subgroup of order p and (a year before that) with the general study of the blocks of defect one. Brauer's achievements are of great importance for character theory and have many consequences.

Many years later (in 1967), J. Thompson used Green's new results on indecomposable modules to produce another proof of some of Brauer's earlier theorems. Afterwards, E. C. Dade was able to give a full description of the blocks with cyclic defect groups.

Here, we will restrict ourselves to the analysis of the p-blocks of the groups having a Sylow p-subgroup of order p; probably, one of the most important cases. To do that, we will need to study the fundamental paper of Brauer on blocks of defect one. (After this is done, it is not very difficult to give a complete description of the blocks of defect one.) Brauer's results, however, cannot be obtained by purely character theoretic methods. We will need to use representations, Schur indices and some algebraic number theory.

We start with the statement for the principal block case (so that the reader can immediately see how powerful these results are).