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.).

On the classification of varieties generated by wreath products of groups

“…Notice that (3.1) and (3.2) both consist of three summands of which the first and the third are the same. An easy comparison of the second summands (see [19] for details) shows that the nilpotency class of the t-generator groupà Wr Y (z, t) from the variety var(A) var(B) is higher than the maximum of the nilpotency classes of the t-generator groups in var(A Wr B) for all large enough t. Thusà Wr Y (z, t) ∈ var(A Wr B).…”

“…The notations and technique with K p -series in this section are just for the first proof only, and they are not needed for understanding the rest of this paper. So we recommend not to focus on them and skip to Lemma 3.1 unless the reader is interested to see our application of Shield's formula to varieties of groups (see details in [19]).…”

“…Ifà is a z-generator group, denote Y (z, t) = C t−z p v . We proved in [19] that there is a positive integer t 1 such that for all t > t 1 the nilpotency class of the wreath productà Wr Y (z, t) is equal to:…”

“…Technique of the arguments below is based both on traditional theory of varieties of groups and on some newer approaches. In Section 3 we suggest the idea of application of K p -series and of D. Shield's formula [26,27] as tools to describe some non-nilpotent varieties via their finitely-generated nilpotent groups (see more details in [19]). Also, our usage of Fitting and Frattini subgroups is based not only on their well known properties reflected, for example, in [21,25] but also on initial articles of W. Gaschütz and H. Fitting [7,8,6]: in their original work some results are formulated in sharper forms for certain specific cases, which are relevant in our study (compare constructions with critical groups and the application of Theorem of Clifford in Section 2 below with constructions in [21, Section 2 in Chapter 5] or with [4]).…”

“…Ol'shanskii, who suggested a shorter proof for Lemma 3.1 by the arguments of [24]. In Section 3 we present both proofs, and the most part of considerations with K p -series is not included into the current text, we just provide references to [19] where closer discussion on K p -series is presented.…”