Geodesic
The criterion of Shmel'kin and varieties generated by wreath products of finite groups
Algebra i logika, Tome 56 (2017) no. 2, pp. 164-175

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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.
Keywords: wreath products, varieties of groups, finite groups, products of varieties of groups, Abelian groups, nilpotent groups, critical groups, Cross varieties. 
@article{AL_2017_56_2_a2,
     author = {V. H. Mikaelian},
     title = {The criterion of {Shmel'kin} and varieties generated by wreath products of finite groups},
     journal = {Algebra i logika},
     pages = {164--175},
     publisher = {mathdoc},
     volume = {56},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2017_56_2_a2/}
}
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V. H. Mikaelian. The criterion of Shmel'kin and varieties generated by wreath products of finite groups. Algebra i logika, Tome 56 (2017) no. 2, pp. 164-175. http://geodesic.mathdoc.fr/item/AL_2017_56_2_a2/