The criterion of Shmel'kin and varieties generated by wreath products of finite groups

We present a general criterion under which the equality $\operatorname{var}(A\operatorname{wr}B)=\operatorname{var}(A)\operatorname{var}(B)$ holds for finite groups $A$ and $B$. This generalizes some known results in this direction and continues our previous research [J. Algebra, 313, No. 2 (2007), 455–458] on varieties generated by wreath products of Abelian groups. The classification is based on the techniques developed by A. L. Shmel'kin, R. Burns, etc., who used critical groups, verbal wreath products, and Cross properties for studying critical groups in nilpotent-by-Abelian varieties.