A Classification Theorem for Varieties Generated by Wreath Products of   Groups
  We suggest a criterion under which for a nilpotent group of finite exponent $A$ and for an abelian group $B$ the variety $var(A \,Wr\, B)$ generated by their wreath product $A \,Wr\, B$ is equal to the product of varieties $var(A)$ and $var(B)$ generated by $A$ and $B$. Namely the equality holds if and only if either the group $B$ is not of some non-zero exponent; or if $B$ is of a non-zero exponent $n$, and $B$ contains a subgroup isomorphic to $C_{d}^c \times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, $d$ is the largest divisor of $n$ coprime with $m$, $C_{d}^c$ is the direct power of $c$ copies of the cycle $C_d$ of order $d$, $C_{n/d}^\infty$ is the direct power of countably many copies of the cycle $C_{n/d}$ of order $n/d$. This criterion continues our previous work on cases when the similar criterions were given for wreath products of abelian groups or of finite groups. Also, this is a generalization of known results in literature, which solve the same problem for much more restricted cases. Some applications of the criterion are considered at the end of paper. 
Vahagn H. Mikaelian
math.GR