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.