A characterisation of nilpotent blocks
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- by Radha Kessar, Markus Linckelmann and Gabriel Navarro PDF
- Proc. Amer. Math. Soc. 143 (2015), 5129-5138 Request permission
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 $|G:P|^2|P:R|$, 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.References
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Additional Information
- Radha Kessar
- Affiliation: Department of Mathematics, City University, London EC1V 0HB, Great Britain
- MR Author ID: 614227
- Email: radha.kessar.1@city.ac.uk
- Markus Linckelmann
- Affiliation: Department of Mathematics, City University, London EC1V 0HB, Great Britain
- MR Author ID: 240411
- Email: markus.linckelmann.1@city.ac.uk
- Gabriel Navarro
- Affiliation: Departament d’Àlgebra, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Spain
- MR Author ID: 129760
- Email: gabriel.navarro@uv.es
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5129-5138
- MSC (2010): Primary 20C20
- DOI: https://doi.org/10.1090/proc/12646
- MathSciNet review: 3411131