= A perfect-base alternating wreath product is generated by elements of order three
For any <perfect group> $H$ and $m\ge3$, the <permutation wreath product> $H\operatorname{wr}A_m$ is generated by elements of order three. The top $A_m$ is generated by 3-cycles. With $\sigma=(1\,2\,3)$, the product of the conjugate $h_1\sigma h_1^{-1}$ and $\sigma^{-1}$ is $h_1h_2^{-1}$; using $\sigma^{-1}$ gives $h_1h_3^{-1}$. The commutator of $h_1h_2^{-1}$ and $k_1k_3^{-1}$ is $[h,k]_1$, since distinct coordinate groups commute. Perfectness gives the entire first coordinate group, and transitivity gives all coordinates. No finite-generation assumption on $H$ is needed.
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