Finite Groups with Some Pronormal Subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to Hx in （H, Hx）. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups （non-PRN-group all of whose proper subgroups are PRN-groups）. At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.