Loewy lengths of blocks with abelian defect groups
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- by Charles W. Eaton and Michael Livesey HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 4 (2017), 21-30
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 $|D|=2^d$. 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 $|D|$ 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.References
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Additional Information
- Charles W. Eaton
- Affiliation: School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
- MR Author ID: 661066
- Email: charles.eaton@manchester.ac.uk
- Michael Livesey
- Affiliation: School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom
- MR Author ID: 1105808
- Email: michael.livesey@manchester.ac.uk
- 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
- © Copyright 2017 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 4 (2017), 21-30
- MSC (2010): Primary 20C20
- DOI: https://doi.org/10.1090/bproc/28
- MathSciNet review: 3682626