  In this paper a study of $G$-minimality, i.e., minimality of four-manifolds
equipped with an action of a finite group $G$, is initiated. We focus on cyclic
actions on $CP^2\# \overline{CP^2}$, and our work shows that even in this
simple setting, the comparison of $G$-minimality in the various categories,
i.e., locally linear, smooth, and symplectic, is already delicate and
interesting. For example, we show that if a symplectic $Z_n$-action on $CP^2\#
\overline{CP^2}$ has an invariant locally linear topological $(-1)$-sphere,
then it must admit an invariant symplectic $(-1)$-sphere, provided that $n=2$
or $n$ is odd. For the case where $n>2$ and even, the same conclusion holds
under a stronger assumption, i.e., the invariant $(-1)$-sphere is smoothly
embedded. Along the way of these proofs we develop certain techniques for
producing embedded invariant $J$-holomorphic two-spheres of self-intersection
$-r$ under a weaker assumption of an invariant smooth $(-r)$-sphere for $r$
relatively small compared with the group order $n$. We then apply the
techniques to give a classification of $G$-Hirzebruch surfaces (i.e.,
Hirzebruch surfaces equipped with a homologically trivial, holomorphic
$G=Z_n$-action) up to orientation-preserving equivariant diffeomorphisms. The
main issue of the classification is to distinguish non-diffeomorphic
$G$-Hirzebruch surfaces which have the same fixed-point set structure. An
interesting discovery is that these non-diffeomorphic $G$-Hirzebruch surfaces
have distinct equivariant Gromov-Taubes invariant, giving the first examples of
such kind. Going back to the original question of $G$-minimality, we show that
for $G=Z_n$, a minimal rational $G$-surface is minimal as a symplectic
$G$-manifold if and only if it is minimal as a smooth $G$-manifold.
