For any group G and any commutative ring R, the Gorenstein homological dimension GhdRG, defined as the Gorenstein flat dimension of the trivial RG-module R, is characterized. We prove that GhdRG<∞ if and only if Gorenstein flat dimension of any RG-module is finite, whenever the supremum of flat dimension of injective modules sfliR over the coefficient ring R is finite. As applications, properties of Ghd on subgroup, quotient group, extension of groups as well as Weyl group are investigated, and moreover, we compare the relations between some invariants such as sfliRG, silfRG, spliRG, silpRG, and Gorenstein projective, Gorenstein flat and PGF dimensions of modules over group rings RG.