University of Isfahan International Journal of Group Theory 2251-7650 13 2 2024 06 01 Structure of finite groups with trait of non-normal subgroups II 173 188 27804 10.22108/ijgt.2023.135678.1814 EN Hamid Mousavi Department of Mathematics, University of Tabriz, P.O.Box 51666-17766, Tabriz, Iran Journal Article 2022 11 09 A finite non-Dedekind group $G$ is called an 𝒩𝒜𝒞-group if all non-normal abelian subgroups are cyclic. In this paper, all finite 𝒩𝒜𝒞-groups will be characterized. Also, it will be shown that the center of non-nilpotent 𝒩𝒜𝒞- groups is cyclic. If 𝒩𝒜𝒞-group $G$ has a non-abelian non-normal Sylow subgroup of odd order, then other Sylow subgroups of $G$ are cyclic or of quaternion type. https://ijgt.ui.ac.ir/article_27804_29287c90e76aaf20667cf7264d270db5.pdf