Skip to main content Accessibility help
×
Home

ON A LATTICE CHARACTERISATION OF FINITE SOLUBLE PST-GROUPS

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

Abstract

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.

Copyright

Corresponding author

Footnotes

Hide All

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

Footnotes

References

Hide All
[1] Ballester-Bolinches, A., Doerk, K. and Pèrez-Ramos, M. D., ‘On the lattice of 𝔉-subnormal subgroups’, J. Algebra 148 (1992), 4252.
[2] Ballester-Bolinches, A. and Esteban-Romero, R., ‘Sylow permutable subnormal subgroups of finite groups II’, Bull. Aust. Math. Soc. 64 (2001), 479486.
[3] Ballester-Bolinches, A., Esteban-Romero, R. and Asaad, M., Products of Finite Groups (Walter de Gruyter, Berlin–New York, 2010).
[4] Ballester-Bolinches, A. and Ezquerro, L. M., Classes of Finite Groups (Springer, Dordrecht, 2006).
[5] Doerk, K. and Hawkes, T., Finite Soluble Groups (Walter de Gruyter, Berlin–New York, 1992).
[6] Kegel, O., ‘Sylow-Gruppen and Subnormalteiler endlicher Gruppen’, Math. Z. 78 (1962), 205221.
[7] Kegel, O. H., ‘Untergruppenverbände endlicher Gruppen, die den Subnormalteilerverband echt enthalten’, Arch. Math. 30(3) (1978), 225228.
[8] Schmidt, R., Subgroup Lattices of Groups (Walter de Gruyter, Berlin, 1994).
[9] Shemetkov, L. A. and Skiba, A. N., Formations of Algebraic Systems (Nauka, Moscow, 1989).
[10] Skiba, A. N., ‘On 𝜎-subnormal and 𝜎-permutable subgroups of finite groups’, J. Algebra 436 (2015), 116.
[11] Vasil’ev, A. F., Kamornikov, A. F. and Semenchuk, V. N., ‘On lattices of subgroups of finite groups’, in: Infinite Groups and Related Algebraic Structures (ed. Chernikov, N. S.) (Institut Matematiki AN Ukrainy, Kiev, 1993), 2754 (in Russian).
[12] Wielandt, H., ‘Eine Verallgemeinerung der invarianten Untergruppen’, Math. Z. 45 (1939), 200244.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed