Brou\'e's Conjecture for 2-blocks with elementary abelian defect groups of order 32

TL;DR

This paper completes the classification of certain 2-blocks with elementary abelian defect groups of order 32, confirming Broué's Abelian Defect Group Conjecture in these cases by analyzing Morita equivalence and lower defect groups.

Contribution

It finalizes the classification of these blocks and verifies Broué's conjecture for them, extending previous partial results.

Findings

Classification of 2-blocks with elementary abelian defect group of order 32

Verification of Broué's Abelian Defect Group Conjecture in these cases

Morita equivalence class determines the inertial quotient in all but three cases

Abstract

The first author has recently classified the Morita equivalence classes of 2-blocks B of finite groups with elementary abelian defect group of order 32. In all but three cases he proved that the Morita equivalence class determines the inertial quotient of B. We finish the remaining cases by utilizing the theory of lower defect groups. As a corollary, we verify Brou\'e's Abelian Defect Group Conjecture in this situation.