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
MathSciNet review:
3682626
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Abstract | References | Similar Articles | Additional Information
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.
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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
Article copyright:
© Copyright 2017
by the authors under
Creative Commons Attribution 3.0 License
(CC BY 3.0)