Finite groups with 𝑂𝑆-propermutable subgroups
A subgroup of a group is called 𝑂𝑆-propermutable in if there is a subgroup such that = 𝑁𝐺(𝐴)𝐵, is a subgroup of and the subgroup permutes with all Schmidt subgroups of 𝐵. In this situation, the subgroup is called 𝑂𝑆-prosupplement to in 𝐺. In this paper, we proved the 𝑝-solubility of a finite group such that a Sylow 𝑝-subgroup of is 𝑂𝑆-propermutable in 𝐺, where > 5.