On the classification of varieties generated by wreath products of groups

Abstract

We suggest a criterion classifying all the cases when for a nilpotent group A of a restricted exponent and for any abelian group B the variety var(A Wr B) generated by the wreath product A Wr B is equal to the product variety var(A) var(B). This continues our previous research on varieties generated by wreath products of other classes of groups (abelian groups, finite groups, etc.). The obtained theorem generalizes some known results in the literature considering the same problem for more restricted cases. Some applications of the criterion also are discussed.

Subvariety structures in certain product varieties of groups

“…For chronological development of this approach and for background information on varieties of groups or on wreath products we refer to [20,21,8,2,3,25,6] and to literature cited therein. Generalizing some results in the cited literature, we in [13]- [18] were able to suggest criteria classifying all the cases when ( * ) holds for groups from certain classes of groups: abelian groups, p-groups, nilpotent groups of finite exponent, etc. (see, in particular, very brief outline of results in Section 5 of [18]).…”

“…Generalizing some results in the cited literature, we in [13]- [18] were able to suggest criteria classifying all the cases when ( * ) holds for groups from certain classes of groups: abelian groups, p-groups, nilpotent groups of finite exponent, etc. (see, in particular, very brief outline of results in Section 5 of [18]).…”

“…Denote by p u the highest power of p dividing n (in terms of (2.2) we could write u = u 1 ). The following technical fact was proved in [18]: The analog of this lemma holds for p-groups in variety var (AWr B 2 ) also. In order to prove the necessity part of Theorem 2.3 (supposing by our agreement made after Lemma 3.1 that A 1 = A 2 = A), assume the equality (3.2) holds for A, B 1 , B 2 .…”

“…Denote by p u the highest power of p dividing n (in terms of (2.2) we could write u = u 1 ). The following technical fact was proved in [18]: Lemma 3.8 (see the proof of Lemma 3.3 in [18]). Any p-group in variety var (AWr B 1 ) belongs to var S p Wr B 1 (p) .…”