%0 Journal Article
%T On some groups whose subnormal subgroups are contranormal-free
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Kurdachenko, Leonid A.
%A Longobardi, Patrizia
%A Maj, Mercede
%D 2024
%\ 05/05/2024
%V
%N
%P -
%! On some groups whose subnormal subgroups are contranormal-free
%K contranormal subgroups
%K subnormal subgroups
%K Nilpotent groups
%K hypercentral groups
%K upper central series
%R 10.22108/ijgt.2024.139136.1871
%X 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.
%U https://ijgt.ui.ac.ir/article_28378_bec3ddac3956ad7f8cf5f6aed206c552.pdf