On $K_p$-series and varieties generated by wreath products of $p$-groups

Abstract

Let $A$ be a nilpotent $p$-group of finite exponent, and $B$ be an abelian $p$-groups of finite exponent. Then the wreath product $A\mathrm{Wr}B$ generates the variety $\mathrm{var}(A)\mathrm{var}(B)$ if and only if the group $B$ contains a subgroup isomorphic to the direct product $C_{p^v}^{\infty }$ of at least countably many copies of the cyclic group $C_{p^v}$ of order $p^v=\exp (B)$. The obtained theorem continues our previous study of cases when $\mathrm{var}(A\mathrm{Wr}B)=\mathrm{var}(A)\mathrm{var}(B)$ holds for some other classes of groups $A$ and $B$ (abelian groups, finite groups, etc.).