A finite group G is called an F-group if for every implies that On the other hand, two elements of a group are said to be z-equivalent or in the same z-class if their centralizers are conjugate in the group. In this paper, for a finite non-abelian group, we give a necessary and sufficient condition for the number of centralizers/z-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of z-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and z-classes of some finite groups and extend some previous results.