University of IsfahanInternational Journal of Group Theory2251-765020240505On some groups whose subnormal subgroups are contranormal-free2837810.22108/ijgt.2024.139136.1871ENLeonid A.KurdachenkoDepartment of Algebra and Geometry, School of Mathematics and Mechanics, National Dnipro University, Gagarin
Prospect 72, Dnipro 10, 49010 Ukraine0000-0002-6368-7319PatriziaLongobardiDepartment of Mathematics, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy0000-0002-7153-5243MercedeMajDepartment of Mathematics, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy0000-0001-7010-5391Journal Article20230915If $G$ 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.https://ijgt.ui.ac.ir/article_28378_bec3ddac3956ad7f8cf5f6aed206c552.pdf