Varieties Generated by Wreath Products of Abelian and Nilpotent Groups

Abstract

We present the full classi cation of all cases, when the (cartesian or direct) wreath product of any abelian groups A and B generates the product variety var (A) var (B). We also present some results on cases when the similar question is considered for non-abelian groups. Our aim is to present our research of recent years on varieties, generated by wreath products of abelian groups and of sets of abelian groups [3] [6], and to display some newest, yet unpublished progress concerning wreath products of non-abelian groups also. In particular, we give complete classi cation of all cases when for the abelian groups A and B their cartesian (or direct) wreath product generates the variety var (A) var (B). The notation Wr below means (standard) cartesian wreath product, but all the statements hold true for direct wreath products also. A product UV of varieties U and V is de ned as the variety of extensions of all groups A ∈ U by all groups B ∈ V. By the Kaloujnine-Krassner theorem extensions of A by B can be embedded into the cartesian wreath product AWrB. Take A and B to be some xed groups, generating the varieties U and V respectively. If var (AWrB) = UV, (1) then it is easier to consider var (AWrB), rather than to study all the extensions in UV. Examples, when this approach is used, are too many to list (see [7]). ∗The author was supported in part by joint grant 13RF-030 of RFBR and SCS MES RA, and by 13-1A246 grant of SCS MES RA.