  A finite group $G$ is called an F-group if for every $x, y \in G \setminus
Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, 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
group, we give necessary and sufficient conditions 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.
