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

TL;DR

This paper characterizes when the wreath product of certain $p$-groups generates a specific variety, linking it to the subgroup structure of the abelian group involved.

Contribution

It provides a precise criterion involving subgroup structure for the wreath product of nilpotent and abelian $p$-groups to generate a particular variety.

Findings

Wreath product generates the variety if and only if the abelian group contains a large subgroup of cyclic groups.

The result extends previous work on varieties generated by wreath products of various classes of groups.

The criterion involves the presence of a subgroup isomorphic to an infinite direct product of cyclic 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 $A {\rm Wr} B$ generates the variety ${\rm var}(A) {\rm var}(B)$ 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 ${\rm var}(A {\rm Wr} B ) = {\rm var}(A){\rm var}(B)$ holds for some other classes of groups $A$ and $B$ (abelian groups, finite groups, etc.).