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

Let A be a nilpotent p-group of finite exponent and B be an abelian p-group of finite exponent for a given prime number p. Then the wreath product A Wr B generates the variety var(A) var(B) if and only if the group B contains a subgroup isomorphic to the direct product C-p upsilon(infinity) of countably many copies of the cycle C-p upsilon of order p(upsilon) = exp(B). The obtained theorem continues our previous study of cases when var(AWrB) = var(A) var(B) holds for some other classes of groups A and B (abelian groups, finite groups, etc.).