Articles written in Proceedings – Mathematical Sciences
Volume 124 Issue 4 November 2014 pp 511-516
Let 𝐺 be a finite group. A subgroup 𝐻 of 𝐺 is called 𝑠-permutable in 𝐺 if it permutes with every Sylow subgroup of 𝐺, and 𝐺 is called a $PS$-group if all minimal subgroups and cyclic subgroups with order 4 of 𝐺 are 𝑠-permutable in 𝐺. In this paper, we give a complete classification of finite groups which are not $PS$-groups but their proper subgroups are all $PS$-groups.