Dihedral defect groups

Abstract

This paper determines much of the structure of blocks whose defect group is dihedral, semidihedral or generalised quaternion and which have either one or two simple modular representations (Brauer characters). It is shown that in the above circumstances there is only a very small number of possibilities for the Cartan matrix, decomposition matrix and the category of modular representations once the defect group is specified. Certain character-theoretic results of Brauer and Olsson are complemented here by a classification of all symmetric algebras with the appropriate representation theory. It is likely that a similar approach is available in the case of blocks with such defect groups and three (the maximum number possible) Brauer characters. In view of the length of such a project this case is being published first. Although the structure theorem can be expressed analogously to that for cyclic defect groups, the inductive proof used there apparently is not applicable here.