2-blocks with abelian defect groups

TL;DR

This paper classifies 2-blocks with abelian defect groups up to Morita equivalence, confirming Donovan's conjecture for certain cases and bounding Cartan matrix entries, using the classification of finite simple groups.

Contribution

It provides a complete Morita classification of 2-blocks with abelian defect groups, including those of rank 2 and order 16, and confirms Donovan's conjecture in these cases.

Findings

Donovan's conjecture holds for elementary abelian 2-groups.

Cartan matrix entries are bounded in terms of defect for abelian 2-groups.

2-blocks with defect groups of the form C_{2^m} x C_{2^m} have two Morita types.

Abstract

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form $C_{2^m} \times C_{2^m}$ for $m \geq 2$ has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita equivalence types of 2-blocks with abelian defect groups of rank 2, from which we conclude that Donovan's conjecture holds for such 2-groups. A further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16. The proof uses the classification of finite simple groups.