A generalisation of Schenkman's theorem

Abstract

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 $S$.