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ON A LATTICE CHARACTERISATION OF FINITE SOLUBLE PST-GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 10 July 2019, pp. 247-254
- Print publication:
- April 2020
-
- Article
- Export citation
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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.
