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On Loewy lengths of blocks

Published online by Cambridge University Press:  20 February 2014

SHIGEO KOSHITANI ,
BURKHARD KÜLSHAMMER  and
BENJAMIN SAMBALE
SHIGEO KOSHITANI
Affiliation:
Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522Japan. e-mail: [email protected]
BURKHARD KÜLSHAMMER
Affiliation:
Mathematisches Institut, Friedrich–Schiller–Universität, D-07737 Jena, Germany. e-mail: [email protected]
BENJAMIN SAMBALE
Affiliation:
Mathematisches Institut, Friedrich–Schiller–Universität, D-07737 Jena, Germany. e-mail: [email protected]
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 3 , May 2014 , pp. 555 - 570
Copyright
Copyright © Cambridge Philosophical Society 2014 

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