On identities in the products of group varieties

Let ℬn be the variety of groups satisfying the identity xn = 1. It is proved that for every sufficiently large prime p, say p > 1010, the product ℬpℬp cannot be defined by a finite set of identities. This solves the problem formulated by C. K. Gupta and A. N. Krasilnikov in 2003. We also find the axiomatic and the basis ranks of the variety ℬpℬp. For this goal, we improve the estimate for the basis rank of the product of group varieties obtained by G. Baumslag, B. H. Neumann, H. Neumann and P. M. Neumann in 1964.