申克曼定理的推广
A generalisation of Schenkman's theorem
摘要：
设$G$是有限群，$\mathfrak{F}$是遗传饱和群系.我们用$\mathbf{Z}_{\mathfrak{F}}(G)$表示$G$的所有正常子群$N$的乘积，使得$G$的每个低于$N$的主因子$H/K$是$G$中的$\mathfrak{F}$ -中心，即\[ (H/K) \rtimes (G/\mathbf{C}_{G}(H/K)) \in \mathfrak{F}. \]。子群$A \leq G$被称为Kegel意义上的$\mathfrak{F}$ -亚正常，或$G$中的$K$ - $\mathfrak{F}$ -亚正常，如果存在子组链\[ A = A_0 \leq A_1 \leq \ldots \leq A_n = G \]，使得对于所有$i = 1, \ldots , n$都是$A_{i-1} \trianglelefteq A_{i}$或$A_i / (A_{i-1})_{A_i} \in \mathfrak{F}$。本文证明了Schenkman关于次正规子群的幂零剩余的中心化子定理的如下推广：设$\mathfrak{F}$是遗传饱和群系，$S$是$G$的$K$ - $\mathfrak{F}$ -次正规子群.如果对于$G$的每个子组$E$，有$\mathbf{Z}_{\mathfrak{F}}(E) = 1$，使得$S \leq E$，则有$\mathbf{C}_{G}(D) \leq D$，其中$D = S^{\mathfrak{F}}$是$S$的$\mathfrak{F}$ -残差。
英文摘要：
Let $G$ be a finite group and let $\mathfrak{F}$ be a hereditary saturated formation. We denote by $\mathbf{Z}_{\mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $\mathfrak{F}$-central in $G$, that is, \[ (H/K) \rtimes (G/\mathbf{C}_{G}(H/K)) \in \mathfrak{F}. \]A subgroup $A \leq G$ is said to be $\mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$\mathfrak{F}$-subnormal in $G$, if there is a subgroup chain \[ A = A_0 \leq A_1 \leq \ldots \leq A_n = G \] such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_i / (A_{i-1})_{A_i} \in \mathfrak{F}$ for all $i = 1, \ldots , n$. In this paper, we prove the following generalisation of Schenkman's Theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let $\mathfrak{F}$ be a hereditary saturated formation and let $S$ be a $K$-$\mathfrak{F}$-subnormal subgroup of $G$. If $\mathbf{Z}_{\mathfrak{F}}(E) = 1$ for every subgroup $E$ of $G$ such that $S \leq E$ then $\mathbf{C}_{G}(D) \leq D$, where $D = S^{\mathfrak{F}}$ is the $\mathfrak{F}$-residual of $S$.
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