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A characterisation of nilpotent blocks


Authors: Radha Kessar, Markus Linckelmann and Gabriel Navarro
Journal: Proc. Amer. Math. Soc. 143 (2015), 5129-5138
MSC (2010): Primary 20C20
DOI: https://doi.org/10.1090/proc/12646
Published electronically: June 30, 2015
MathSciNet review: 3411131
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Abstract: Let $ B$ be a $ p$-block of a finite group, and set $ m=$ $ \sum \chi (1)^2$, the sum taken over all height zero characters of $ B$. Motivated by a result of M. Isaacs characterising $ p$-nilpotent finite groups in terms of character degrees, we show that $ B$ is nilpotent if and only if the exact power of $ p$ dividing $ m$ is equal to the $ p$-part of $ \vert G:P\vert^2\vert P:R\vert$, where $ P$ is a defect group of $ B$ and where $ R$ is the focal subgroup of $ P$ with respect to a fusion system $ \mathcal {F}$ of $ B$ on $ P$. The proof involves the hyperfocal subalgebra $ D$ of a source algebra of $ B$. We conjecture that all ordinary irreducible characters of $ D$ have degree prime to $ p$ if and only if the $ \mathcal {F}$-hyperfocal subgroup of $ P$ is abelian.


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Additional Information

Radha Kessar
Affiliation: Department of Mathematics, City University, London EC1V 0HB, Great Britain
Email: radha.kessar.1@city.ac.uk

Markus Linckelmann
Affiliation: Department of Mathematics, City University, London EC1V 0HB, Great Britain
Email: markus.linckelmann.1@city.ac.uk

Gabriel Navarro
Affiliation: Departament d’Àlgebra, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Spain
Email: gabriel.navarro@uv.es

DOI: https://doi.org/10.1090/proc/12646
Keywords: Nilpotent block, height zero, hyperfocal subalgebra
Received by editor(s): February 24, 2014
Received by editor(s) in revised form: July 2, 2014, and September 30, 2014
Published electronically: June 30, 2015
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society