Let Bnβ be the variety of groups satisfying the law xn=1. It is
proved that for every sufficiently large prime p, say p>1010, the
product BpβBpβ 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 BpβBpβ. 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 long ago.Comment: 9 page