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2-blocks with abelian defect groups

Eaton, C.W., Kessar, R., Kuelshammer, B. & Sambale, B. (2014). 2-blocks with abelian defect groups. ADVANCES IN MATHEMATICS, 254, pp. 706-735. doi: 10.1016/j.aim.2013.12.024


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 Donovanʼs 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 C2m×C2m for m⩾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 Donovanʼs 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.

Publication Type: Article
Additional Information: NOTICE: this is the author’s version of a work that was accepted for publication in ADVANCES IN MATHEMATICS. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in ADVANCES IN MATHEMATICS, Vol.254, (14.03.2014) 10.1016/j.aim.2013.12.024
Publisher Keywords: Block; Defect; Character; Donovanʼs conjecture; Cartan invariant; Morita equivalence; Brauer correspondence
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
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