Graded polynomial identities of matrices

We consider G-graded polynomial identities of the p x p matrix algebra M-p(K) over a field K of characteristic 0 graded by an arbitrary group G. We find relations between the G-graded identities of the G-graded algebra M-p(K) and the (G x H)-graded identities of the tensor product of M-p(K) and the H-graded algebra M-q (K) with a fine H-grading. We also find a basis of the G-graded identities of M-p(K) with an elementary grading such that the identity component coincides with the diagonal of M-p(K) (C) 2002 Elsevier Science Inc. All rights reserved.