Abstract: The necessary and sufficient condition for finite groups to be p-nilpotent groups is given by using c-normality,and two sufficient conditions for finite groups G to be p-supersolvable groups are obtained by investigating the c-normality of Sylow subgroups and their maximal subgroups.

Key words: c-normal subgroups, p-nilpotent groups, p-solvable groups, p-supersolvable groups