Splendid Morita equivalences for principal blocks with generalised quaternion defect groups

TL;DR

This paper demonstrates how splendid Morita equivalences between principal blocks with dihedral Sylow 2-subgroups can be extended to those with generalized quaternion Sylow 2-subgroups, using Scott modules.

Contribution

It introduces a method to lift Morita equivalences from dihedral to generalized quaternion Sylow 2-subgroups via Scott modules.

Findings

Morita equivalences can be lifted between these blocks.

Scott modules play a key role in the lifting process.

The results unify understanding of block equivalences for different Sylow subgroup types.

Abstract

We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow $2$-subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow $2$-subgroups realised by Scott modules.