Morita Equivalence Classes of Tame Blocks of Finite Groups

TL;DR

This paper refines the classification of blocks of finite groups by showing certain Morita equivalence classes of tame algebras do not occur in this context, providing a complete classification for dihedral defect groups.

Contribution

It establishes which Morita equivalence classes of tame algebras can occur as blocks of finite groups, completing the classification for dihedral defect groups.

Findings

Certain Morita classes do not occur as blocks of finite groups

Complete classification for blocks with dihedral defect groups

Refinement of Erdmann's classifications

Abstract

We show that several Morita equivalence classes of tame algebras do not occur as blocks of finite groups. This refines classifications by Erdmann of classes of blocks with dihedral, semidihedral, and generalised quaternion defect groups. In particular we now have a complete classification of the Morita equivalence classes of blocks of finite groups with dihedral defect groups.