# View a group acting faithfully and transitively on a set $X$ as a subgroup of a wreath product.

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.

The action of $G$ in $X$ not need to be faithful. This result is the "Embedding Theorem" and can be found -along with the proof- in the monograph "Wreath product of groups and semigroups", Meldrum 1995, Theorem $2.6$.
For anyone interested about, this situation applies in the following context. Let $G$ be a finite group and $V$ a finite $G$-module of positive characteristic. Le $N$ normal in $G$, then, if $V$ is faithful and irreducible under $G$, by Clifford's Theorem $V_N$ (i.e. $V$ viewed as $N$-module) is completely reducible under $N$ and decomposes in homogeneuos components $V_N=V_1 \oplus \dots \oplus V_s$. Moreover $G/N$ acts transitively on the $V_i$'s. Suppose $V_N$ is not homogeneous. Then $V$, as $G$-set, has $V_1$ as imprimitivity block. Using that $V$ is faithful we have $$core_G(C_G(V_1))=\bigcap_i C_G(V_i) =1$$ Moreover, $C_G(V_i) \trianglelefteq N_G(V_i)$ and the normalizer $N_G(V_i)$ is is the stabilizer of $V_i$ viewed as imprimitivity block. If $H= N_G(V_i)/C_G(V_i)$ and $K=G/core_G(N_G(V_1))$ (this is a group that faithfully permutes the homogeneous components), the result in the question gives us an embedding $$G \lesssim H \wr K$$
• Zeros of Brauer characters and linear actions of finite groups, Dolfi-Pacifici, 2011, remark $2.1$
• Large character degrees of solvable groups with abelian Sylow $2$-subgroups, Dolfi-Jabara, 2006, before paragraph $3$.