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MONOLITHIC BRAUER CHARACTERS
Published online by Cambridge University Press: 28 March 2019
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 $G$.
MSC classification
- 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.
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).
References
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