Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

A subgroup $R$ of a finite group $G$ is weakly subnormal in $G$ if $R$ is not subnormal in $G$ but it is subnormal in every proper overgroup of $R$ in $G$. In this paper, we first classify all finite groups $G$ which contains a weakly subnormal $p$-subgroup for some prime $p$. We then determine all finite groups containing a cyclic weakly subnormal $p$-subgroup. As applications, we prove a number of variations of the Baer-Suzuki theorem using the orders of certain group elements.