The variety generated of the finite group

The purpose of this paper is to give the simple proof of the following theorem (if [I]): the product $\mathfrak{N}\mathfrak{M}$ of the non-trivial varieties $\mathfrak{N}$ and $\mathfrak{M}$ is generated by the finite group if and only if
a) $\mathfrak{N}$ and $\mathfrak{M}$ has non zero coprime exponents and
b) $\mathfrak{N}$ consists of the nilpotent groups and $\mathfrak{M}$ consists of the abelian groups.

References

1. A. L. Šmelkin, The wreath products and the group varieties, Isvestia Akademee Nauk USSR, ser.math., 29,N I (1965), 149–170.