MONOLITHIC BRAUER CHARACTERS
2019 ◽
Vol 100
(3)
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pp. 434-439
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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 $G$.
1991 ◽
Vol 34
(3)
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pp. 423-425
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2015 ◽
Vol 102
(1)
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pp. 96-107
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2001 ◽
Vol 44
(1)
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pp. 111-115
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1974 ◽
Vol 26
(3)
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pp. 746-752
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2017 ◽
Vol 96
(3)
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pp. 426-428
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2013 ◽
Vol 50
(2)
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pp. 258-265
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2018 ◽
Vol 21
(7-8)
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pp. 1515-1517
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