Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$-subgroup of $G$. We prove that $P$ is normal in $G$ if and only if $\unicode[STIX]{x1D711}(1)_{p}^{2}$ divides $|G:\ker (\unicode[STIX]{x1D711})|_{p}$ for all monomial monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

中文翻译：

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BRAUER 特征和正常 SYLOW 亚群的度数

让$p$成为素数，$G$一个可解组和$P$西洛$p$-子群$G$. 我们证明$P$是正常的$G$当且仅当$\unicode[STIX]{x1D711}(1)_{p}^{2}$划分$|G:\ker (\unicode[STIX]{x1D711})|_{p}$对于所有单项式单片不可约$p$-布劳尔角色$\unicode[STIX]{x1D711}$的$G$.