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 $A\le G$ 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 $A_{i-1}⊴A_i$ or $A_i/(A_{i-1}{)}_{A_i}\in \mathfrak{F}$ for all $i=1,\dots ,n$. 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 $S\le E$ then ${\mathbf{C}}_G(D)\le D$, where $D=S^{\mathfrak{F}}$ is the $\mathfrak{F}$-residual of $S$.

9 pages

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