1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $2.6$.

EDIT: It may not have much sense to add some remarks after years, but I hope that this will be useful to someone. The topic is completely described in the chapter 4.4 "Imprimitive Groups and Wreath Products" of "The Theory of Finite Groups: An Introduction", Kurzweil-Stellmacher.

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 \begin{equation} core_G(C_G(V_1))=\bigcap_i C_G(V_i) =1 \end{equation} 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 \begin{equation} G \lesssim H \wr K \end{equation}

This appears often in the study of finite groups and their characters. Some details can be found in

  • 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 $3$.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.