Kp-series and varieties generated by wreath products of p-groups

Abstract

Let [Formula: see text] be a nilpotent [Formula: see text]-group of finite exponent and [Formula: see text] be an abelian [Formula: see text]-group of finite exponent for a given prime number [Formula: see text]. Then the wreath product [Formula: see text] generates the variety [Formula: see text] if and only if the group [Formula: see text] contains a subgroup isomorphic to the direct product [Formula: see text] of countably many copies of the cycle [Formula: see text] of order [Formula: see text]. The obtained theorem continues our previous study of cases when [Formula: see text] holds for some other classes of groups [Formula: see text] and [Formula: see text] (abelian groups, finite groups, etc.).