A subgroup H of a group G is called c-normal in G if there exists a normal subgroup K of G such that \(G=HK\) and \(H\cap K\le H_G\) , where \(H_{G}\) is core of H in G. We present a new criterion for p-supersolvability of finite groups by use of a small quantity of c-normal maximal subgroups of a Sylow p-subgroup. As applications, we obtain some sufficient conditions for a finite group to be p-nilpotent and supersolvable. Some known results are extended.
