  As an instance of a linear action of a Hopf algebra on a free associative
algebra, we consider finite group gradings of a free algebra induced by
gradings on the space spanned by the free generators. The homogeneous component
corresponding to the identity of the group is a free subalgebra which is graded
by the usual degree. We look into its Hilbert series and prove that it is a
rational function by giving an explicit formula. As an application, we show
that, under suitable conditions, this subalgebra is finitely generated if and
only if the grading on the base vector space is trivial.
