I'm studying algebra and I saw that given $H$ group of permutations of a set $\Delta$ and $K$ group of permutation of a set $\Omega$ we have that the wreath product $H\wr K$ is a group of permutations of $\Delta \times \Omega$, moreover the action is not primitive and $\Delta \times \Omega$ admits a non-trivial $(H \wr K)$-invariant partition, namely $\{\Delta \times \{y\} |y \in \Omega\} $.
I'm interested about the converse.
Let be $G$ group acting transitively and faithfully on a set $X$, suppose that the action is not primitive (i.e. exist a non trivial $G$-invariant partition $\Omega$ of $X$). Then there exists a natural transitive action of $G$ over $\Omega$, let $N$ the kernel of this action and set $K=G/N$. Of course, $K$ acts faithfully on $\Omega$.
Now let $\Delta \in \Omega$ and set $H=G_{\Delta}$ the stabilizer of $\Delta$ under the action of $G$ over $\Omega$. Consider $H$ in its action over $\Delta$ (is this action faithful?). Now what I would want to prove is that
Exists a monomorphism $\phi \colon G \to H \wr K$ and, up to identifying $G$ with $G \phi$, $\phi$ can be constructed such that the action of $G$ over $X$ is equivalent of the action of $G$ over $\Delta \times \Omega$. $(*)$
The sense is that a group $G$ acting transitively and faithfully on a set $X$ is, in a certain sense, a subgroup of a wreath product.
The statement above is very useful firstly because I use it later in some statements about the group of automorphims of a rooted tree, and I need to solve dependencies ^^ Secondly because it seems very useful to treat a imprimitive group such as a subgroup of a wreath product.
$(*)$ Let $G$ acting on two set $S$ and $S'$, we say that these two actions are equivalent if there is a biiection $f \colon S \to S'$ such that $f(s \cdot g)=f(s) \cdot g$ for all $g \in G$ and $s \in S$, so we can treat two equivalent action as the same action.
