Let $G$ be a finite group. Then the exponent of $G$ divides $n = p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{r}^{\alpha_{r}}$ ($p_{1} > p_{2} > \cdots > p_{r}$, $p_{i} \text{ odd prime}, \, i = 1, \cdots,r$) and the order of every principal factor of $G$ is $p_{i}$ or $p_{i}^{2}$$(i = 1, \cdots,r)$ if and only if the laws (1), (2), (3) of the Theorem 2.1 are satisfied in $G$. Some varieties generated by groups of this kind are characterized.