S-semipermutable and Weakly S-permutable Subgroups

S-semipermutable subgroup and weakly s-permutable subgroup are two different generalizations of s-permutable subgroup. In this paper, we investigate the influence of s-semipermutable and weakly s-permutable subgroups on the structure of finite groups. We give some conditions of p-nilpotency and supersolvability under assumption that some primary subgroups ( for example, maximal subgroups or minimal subgroups of Sylow subgroups ) are either s-semipermutable or weakly s-permutable. Meanwhile, some results are extended by using formation theory.