Varieties Generated by Wreath Products of Abelian and Nilpotent Groups

Abstract

Presented by the Program Committee of the Conference "Mal'tsev Readings"Our aim is to review recent publications on varieties generated by wreath products of Abelian groups and by sets of Abelian groups [1][2][3][4], and also to present some unpublished facts about wreath products of non-Abelian groups. In particular, we give a complete classification of all cases where for Abelian groups A and B, their Cartesian (or direct) wreath product generates the variety var A var B. Throughout, Wr denotes a (standard) Cartesian wreath product, even though all statements hold true for direct wreath products as well. A product UV of varieties U and V is defined as the variety of all extensions of groups A ∈ U by groups B ∈ V.By the Kaloujnine-Krassner theorem, extensions of A by B can be embedded in the Cartesian wreath product A Wr B. Let A and B be some groups generating varieties U and V, respectively.then it is easier to consider var A Wr B rather than explore all possible extensions in UV. Examples of using this approach are too many to list them (see [5]). A n , where A n is the variety of all Abelian groups of exponent dividing n. C. Houghton generalized this to arbitrary finite cyclic groups A = C m and B = C n . Namely, var C m Wr C n = var C m var C n = A m A n holds iff m and n are coprime (this result is unpublished but is repeatedly mentioned in literature; see, e.g., [5,7]).

On the classification of varieties generated by wreath products of groups

“…Theorem 1.1 continues our research on classification of cases when ( * ) holds for groups A and B of certain classes of groups. In particular, in [15,16] we gave a full classification for ( * ) holding for any abelian groups A and B, and in [17,18] we classified the cases when A and B are any finite groups.…”

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

“…Recall that we above denoted N c,m = N c ∩ B m . In [11], using the properties of critical groups from [3] and Proposition 2 from [11], we saw for the group A = F 2 (N 2,3 ), that (6) var (A wr C 2 ) = var (A) var (C 2 ) = N 2,3 A 2 ,…”

“…Using technics with critical groups in [3, Section 3] one could easily find cases when the equality (1) holds or does not hold, say, for A = F 2 (N 2,p ) and B = C k q , where prime numbers p and q are chosen so that q divides p − 1. Applying methods from our previous research [8]- [11] we generalize this in Theorem 1 for arbitrary finite A and B.…”

“…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. 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.…”