  Let $G$ be a group and $R$ be a ring. We define the Gorenstein homological
dimension of $G$ over $R$, denoted by ${\rm Ghd}_{R}G$, as the Gorenstein flat
dimension of trivial $RG$-module $R$. It is proved that ${\rm Ghd}_SG \leq {\rm
Ghd}_RG$ for any flat extension of commutative rings $R\rightarrow S$; in
particular, ${\rm Ghd}_{R}G$ is a refinement of ${\rm Ghd}_{\mathbb{Z}}G$ if
$R$ is $\mathbb{Z}$-torsion-free. We show a Gorenstein homological version of
Serre's theorem, i.e. ${\rm Ghd}_{R}G = {\rm Ghd}_{R}H$ for any subgroup $H$ of
$G$ with finite index. As an application, $G$ is a finite group if and only if
${\rm Ghd}_{R}G = 0$; this is different from the fact that the homological
dimension of any non-trivial finite group is infinity.
