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  • ZHANG CHI (a1) and ALEXANDER N. SKIBA (a2)


Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$ . A chief factor $H/K$ of $G$ is $\mathfrak{F}$ -central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$ . We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$ -central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$ . As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$ , where $\mathfrak{N}$ is the class of all nilpotent groups.


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Research of the first author is supported by the China Scholarship Council and NNSF of China (11771409).



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