On Loewy lengths of blocks

We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It...

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Bibliographic Details
Published in:Mathematical proceedings of the Cambridge Philosophical Society 2014-05, Vol.156 (3), p.555-570
Main Authors: KOSHITANI, SHIGEO, KÜLSHAMMER, BURKHARD, SAMBALE, BENJAMIN
Format: Article
Language:English
Subjects:
Algebraic group theory
Defects
Group theory
Lists
Lower bounds
Mathematical analysis
Trees
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Summary:We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p=2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004114000103