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MONOLITHIC BRAUER CHARACTERS

Part of: Representation theory of groups

Published online by Cambridge University Press:  28 March 2019

XIAOYOU CHEN  and
MARK L. LEWIS Open the ORCID record for MARK L. LEWIS [Opens in a new window]
XIAOYOU CHEN
Affiliation:
College of Science, Henan University of Technology, Zhengzhou 450001, China email cxymathematics@hotmail.com
MARK L. LEWIS*
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA email lewis@math.kent.edu
*
✉lewis@math.kent.edu
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Abstract

Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

Keywords

Itô–Michler theoremmonolithic Brauer charactersSylow subgroup

MSC classification

Primary: 20C20: Modular representations and characters
Secondary: 20C15: Ordinary representations and characters
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 434 - 439
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

The first author is supported by the Fund for Young Key Teachers of Henan University of Technology, the Fund of Henan Administration of Foreign Experts Affairs, the Project of Henan Province (182102410049) and the NSFC (11571129 and 11771356).

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