  We consider geometrical problems on Gorenstein hypersurface orbifolds of
dimension $n \geq 4$ through the theory of Hilbert scheme of group orbits. For
a linear special group $G$ acting on $\CZ^n$, we study the $G$-Hilbert scheme,
$\hl^G(\CZ^n)$, and crepant resolutions of $\CZ^n/G$ for $G$=the $A$-type
abelian group $ A_r(n)$. For $n=4$, we obtain the explicit structure of
$\hl^{A_r(4)}(\CZ^4)$. The crepant resolutions of $\CZ^4/A_r(4)$ are
constructed through their relation with $\hl^{A_r(4)}(\CZ^4)$, and the
connections between these crepant resolutions are found by the "flop" procedure
of 4-folds. We also make some primitive discussion on $\hl^G(\CZ^n)$ for the
$G$= alternating group ${\goth A}_{n+1}$ of degree $n+1$ with the standard
representation on $\CZ^n$; the detailed structure of $\hl^{{\goth A}_4}(\CZ^3)$
is explicitly constructed.
