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$H$ and $K$ are groups, and $\Gamma$ is a set acted upon by H, while $\Delta$ is a set acted upon by $K$. Let $W := K \wr_\Gamma H$, the wreath product of $H$ and $K$. I have seen theorems stating that, ($H$ on $\Gamma$) and ($K$ on $\Delta$) are both transitive/faithful iff ($W$ on $\Delta \times \Gamma$) is a transitive/faithful action. There are also similar, though not identical results, for the action of $W$ on $\Delta^{\Gamma}$: the set of functions $f: \Gamma \rightarrow \Delta$, based upon the actions of the underlying groups.

Now, here is my question. Let $S$ := $H \ltimes_\phi K$, for some homomorphism $\phi: H \rightarrow \operatorname{Aut}(K)$. Are there similar such results relating the properties of S acting on some set built from $\Delta$ and $\Gamma$, such as their cartesian product, to the properties of ($H$ on $\Gamma$) and ($K$ on $\Delta$)?

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What action do you intend for $S$ on $\Delta \times \Gamma$? I believe the minimum permutation degree of a semi-direct product may exceed the product of the minimum permutation degrees of its components, so certainly the action need not be faithful. I'm not even sure how to make it well defined. – Jack Schmidt Aug 4 '12 at 19:53

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