Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20C20

Retrieve articles in all journals with MSC (2010): 20C20


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

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