  Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)}
and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let
F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F
is a field of zero characteristic. Here the degree of a monomial is determined
by the product of the indices in G. Let I be a G-graded T-ideal of
F<x_{1,g_1},...,x_{r,g_r}> which is PI (e.g. any ideal of identities of a
G-graded finite dimensional algebra is of this type). We prove that the Hilbert
series of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function. More generally,
we show that the Hilbert series which corresponds to any g-homogeneous
component of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function.
