On some groups whose subnormal subgroups are contranormal-free

Kurdachenko, Leonid A.; Longobardi, Patrizia; Maj, Mercede

doi:10.22108/ijgt.2024.139136.1871

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	On some groups whose subnormal subgroups are contranormal-free
International Journal of Group Theory
مقاله 6، دوره 14، شماره 2، شهریور 2025، صفحه 99-115 اصل مقاله (467.46 K)
نوع مقاله: Ischia Group Theory 2022
شناسه دیجیتال (DOI): 10.22108/ijgt.2024.139136.1871
نویسندگان
Leonid A. Kurdachenko¹؛ Patrizia Longobardi²؛ Mercede Maj^* ²
¹Department of Algebra and Geometry, School of Mathematics and Mechanics, National Dnipro University, Gagarin Prospect 72, Dnipro 10, 49010 Ukraine
²Department of Mathematics, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy
چکیده
If $G$ is a group, a subgroup $H$ of $G$ is said to be contranormal in $G$ if $H^G = G$, where $H^G$ is the normal closure of $H$ in $G$. We say that a group is contranormal-free if it does not contain proper contranormal subgroups. Obviously, a nilpotent group is contranormal-free. Conversely, if $G$ is a finite contranormal-free group, then $G$ is nilpotent. We study (infinite) groups whose subnormal subgroups are contranormal-free. We prove that if $G$ is a group which contains a normal nilpotent subgroup $A$ such that $G/A$ is a periodic Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is generated by subnormal nilpotent subgroups; in particular $G$ is a Baer group. Furthermore, if $G$ is a group which contains a normal nilpotent subgroup $A$ such that the $0$-rank of $A$ is finite, the set $\Pi(A)$ is finite, $G/A$ is a Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is a Baer group.
کلیدواژه‌ها
Contranormal subgroups؛ subnormal subgroups؛ nilpotent groups؛ hypercentral groups؛ upper central series

مراجع
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On some groups whose subnormal subgroups are contranormal-free