On the classification of varieties generated by wreath products of groups

We propose a classification of all cases when, for a nilpotent group $A$ of finite exponent and an abelian group $B$, the variety $\operatorname{var}(A \mathbin{\mathrm{Wr}} B)$ generated by their wreath product is equal to the product $\operatorname{var}(A)\operatorname{var}(B)$ of the varieties $\operatorname{var}(A)$ and $\operatorname{var}(B)$ generated by $A$ and $B$. This generalizes some results in the literature concerning the same problem for more restricted types of groups. This classification extends our earlier work on varieties generated by wreath products of abelian groups and wreath products of finite groups.