Metabelian varieties of groups and wreath products of abelian groups

Abstract

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.

On the classification of varieties generated by wreath products of groups

“…The second direction for restriction is to consider Theorem 1.1 for abelian groups. We had proved: Theorem 6.2 (Theorem 6.1 in [15] or Theorem A in [16]). To make this theorem more intact with Theorem 1.1 and Theorem 6.1, let us reformulate it slightly differently.…”

“…Since var(A Wr B) is locally finite, we may assume P is finite. By [21,16.31] var(A Wr B) is generated by all the finitely generated subgroups {R i | i ∈ I} of the group A Wr B, which, clearly, all are finite also.…”

“…To show the opposite inclusion denote π = {p, q} and notice that it will be sufficient to take any π-group P in var(A Wr B), and to show it is in var(S p Wr B(q)). Again, it is enough to consider finite groups P only, and to assume by [21,16.31] that P is a surjective image of a subgroup R of the direct product R 1 × · · · × R l under a homomorphism ρ : R → P for some finite subgroups R i of A Wr B.…”

The Criterion of Shmel’kin and Varieties Generated by Wreath Products of Finite Groups

“…In articles [8]- [11] we presented full classification of all cases when (1) holds for arbitrary abelian groups. In [10] we gave a classification of all cases when the analog of (1) holds for wreath products of sets of abelian groups. After the classification was found for all abelian groups, it is natural to widen the class of groups, and the first class to consider are finite groups.…”

“…After the classification was found for all abelian groups, it is natural to widen the class of groups, and the first class to consider are finite groups. In the listed papers we already had suggested some special cases such as examples 8.5, 8.6 and 8.7 in [10], Proposition 2 and Example 2 in [11] in which the analog of (1) holds or does not hold for some specific non-abelian finite groups.…”