Let (Formula presented.) be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the (Formula presented.)-ideal of (Formula presented.)-graded identities of a (Formula presented.)-graded algebra in characteristic 0 and the generators of the (Formula presented.)-ideal of (Formula presented.)-graded identities of its tensor product by the infinite-dimensional Grassmann algebra (Formula presented.) endowed with the canonical grading have pairly the same degree. In this paper, we deal with (Formula presented.)-graded identities of (Formula presented.) over an infinite field of characteristic (Formula presented.), where (Formula presented.) is (Formula presented.) endowed with a specific (Formula presented.)-grading. We find identities of degree (Formula presented.) and (Formula presented.) while the maximal degree of a generator of the (Formula presented.)-graded identities of (Formula presented.) is (Formula presented.) if (Formula presented.). Moreover, we find a basis of the (Formula presented.)-graded identities of (Formula presented.) and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the (Formula presented.)-graded Gelfand–Kirillov (GK) dimension of (Formula presented.). © 2016 World Scientific Publishing Company26 116