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Proceedings of the American Mathematical Society Series B

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Loewy lengths of blocks with abelian defect groups


Authors: Charles W. Eaton and Michael Livesey
Journal: Proc. Amer. Math. Soc. Ser. B 4 (2017), 21-30
MSC (2010): Primary 20C20
DOI: https://doi.org/10.1090/bproc/28
Published electronically: August 4, 2017
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Abstract: We consider $ p$-blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index $ p$. Using this, we show that if $ B$ is a $ 2$-block of a finite group with abelian defect group $ D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s$, where $ a_i > 1$ for all $ i$ and $ r \geq 0$, then $ d < \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1$, where $ \vert D\vert=2^d$. When $ s=1$ the upper bound can be improved to $ 2^{a_1}+\cdots +2^{a_r}+2-r$. Together these give sharp upper bounds for every isomorphism type of $ D$. A consequence is that when $ D$ is an abelian $ 2$-group the Loewy length is bounded above by $ \vert D\vert$ except when $ D$ is a Klein-four group and $ B$ is Morita equivalent to the principal block of $ A_5$. We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal $ 3$-blocks.


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Additional Information

Charles W. Eaton
Affiliation: School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
Email: charles.eaton@manchester.ac.uk

Michael Livesey
Affiliation: School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
Email: michael.livesey@manchester.ac.uk

DOI: https://doi.org/10.1090/bproc/28
Received by editor(s): July 29, 2016
Received by editor(s) in revised form: November 9, 2016, and November 24, 2016
Published electronically: August 4, 2017
Additional Notes: This research was supported by the EPSRC (grant no. EP/M015548/1).
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2017 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)