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.
Categories: math.AC math.KT