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

Abstract

We present a general criterion under which the equality var(A Wr B) = var(A) var(B) holds for finite groups A and B. This continues our previous research on varieties, generated by wreath products of abelian groups, and generalizes some existing results in this direction in literature. The classification is based on criterion of A.L. Shmel'kin for product varieties of groups and on technics developed by R. Burns et al. on critical groups in nilpotent-by-abelian varieties.