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MONOLITHIC BRAUER CHARACTERS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 3 / December 2019
- Published online by Cambridge University Press:
- 28 March 2019, pp. 434-439
- Print publication:
- December 2019
-
- Article
- Export citation
-
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 
$G$.
