  Let ${\cal B}_n$ be the variety of groups satisfying the law $x^n=1$. It is
proved that for every sufficiently large prime $p$, say $p>10^{10}$, the
product ${\cal B}_p{\cal B}_p$ 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 ${\cal B}_p{\cal
B}_p$. 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.
