Let $ A $ and $ B $ be subgroups in a finite group $ G $ . Then $ A $ is (hereditarily) $ G $ -permutable with $ B $ if $ AB^{x}=B^{x}A $ for some $ x\in G $ (for some $ x\in\langle{}A,B\rangle{} $ ). A subgroup $ A $ in $ G $ is (hereditarily) $ G $ -permutable in $ G $ if $ A $ is (hereditarily) $ G $ -permutable with all subgroupsin $ G $ . The article deals with the structure of $ G $ such thatthe normalizers of Sylow subgroups are (hereditarily) $ G $ -permutable.
