On identities in the products of group varieties

TL;DR

This paper proves that for large primes, the product of certain group varieties cannot be finitely defined, solving a problem posed in 2003, and determines the ranks of these varieties.

Contribution

It demonstrates that for sufficiently large primes, the product of the varieties ${ m f B}_p$ with itself cannot be finitely axiomatized, and calculates their axiomatic and basis ranks.

Findings

The product ${ m f B}_p{ m f B}_p$ is not finitely based for large primes.

The paper determines the axiomatic and basis ranks of ${ m f B}_p{ m f B}_p$.

Improves previous estimates for the basis rank of product varieties.

Abstract

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.