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.