Product varieties of $m$-groups

A new concept of mimicking is introduced. We point out representations that mimic a variety $\mathcal A$ of Abelian $m$-groups and a variety $\mathcal I$ of $m$-groups defined by an identity $x_*=x^{-1}$. It is proved that if a variety $\mathcal U$ of $m$-groups is generated by some class of $m$-groups, and a variety $\mathcal V$ of $m$-groups is mimicked by some class of $m$-groups, then their product $\mathcal{U\cdot V}$ is generated by wreath products of groups in the respective classes. For every natural $n$, we construct $m$-groups generating varieties $\mathcal I_n=(\mathcal I^{n-1})\cdot\mathcal I$ and $\mathcal A_n=(\mathcal A^{n-1})\cdot\mathcal A$.