Weakly subnormal subgroups and variations of the Baer–Suzuki theorem

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