We study the variety generated by cartesian and direct wreath products of arbitrary sets X and Y of abelian groups. In particular, we give a classification of the cases when that variety is equal to the product variety var(X) var(Y). This criterion is a wide generalization of the theorems of Higman and Houghton about the varieties generated by wreath products of cycles, of a few other known examples about the varieties generated by wreath products of abelian groups (and of sets of abelian groups), and also of our recent research about the varieties generated by wreath products of abelian groups.

Abstract. Generalizing results of Higman and Houghton on varieties generated by wreath products of finite cycles, we prove that the (direct or cartesian) wreath product of arbitrary abelian groups A and B generates the product variety var (A) · var (B) if and only if one of the groups A and B is not of finite exponent, or if A and B are of finite exponents m and n respectively and for all primes p dividing both m and n, the factorsand where p k is the highest power of p dividing n.

We present a general criterion under which the equality var (A wr B) = var (A) var (B) holds for finite groups A and B. This generalizes known results in this direction in the literature, and continues our previous research on varieties generated by wreath products of abelian groups. The classification is based on technics developed by A.L. Shmel'kin, R. Burns et al. to study the critical groups in nilpotentby-abelian varieties.

Answering a question of de la Harpe and Bridson in the Kourovka Notebook, we build the explicit embeddings of the additive group of rational numbersℚin a finitely generated groupG. The groupGin fact is two-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubility or torsion freeness.