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I say in advance that I am really new to Group Theory, so if my question is trivial I apologize in advance.

Let $A$ and $H$ be groups and $\Omega$ be a $H$-set. In this set-up, we can define the wreath product $ G: =A\,\text{Wr}_\Omega H$. If $A$ acts on set $\Lambda$, then we get a canonical action of $G$ on the set $\Lambda^\Omega=\lbrace f\colon \Omega \longrightarrow \Lambda| \: f \textit{ is a function}\rbrace$. Wikipedia calls this action the primitive action of the wreath product. My question is the following:

Q: Is the action above indeed primitive? If it is not, do we have to assume something more on the action of $A$ on $\Lambda$ ( for example primitivity) to ensure the primitivity of the action of the wreath product?

Any help or reference is well accepted. Thank you.

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  • $\begingroup$ It may not be primitive, as $A$ could be imprimitive (e.g. $A=C_4$). What about checking the O'Nan-Scott theorem? $\endgroup$ – LeechLattice Sep 17 '19 at 1:13
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    $\begingroup$ Even if $A$ and $H$ are primitive, the wreath product may not be primitive: Let $A=C_3$ and $H=S_3$, both on $3$ points. The wreath product is a subgroup of $AGL(3,3)$ and preserves subspaces of the form $(a,b,c)+(1,1,1)t$ for $t\in \mathbb{Z}_3$. $\endgroup$ – LeechLattice Sep 17 '19 at 3:43
  • $\begingroup$ Thanks for the very helpful comments. If I take the wreath product of $S_n$ and $S_k$ do you think the action on $\lbrace 1,\ldots,n\rbrace ^k$ is still not primitive? $\endgroup$ – Vincenzo Zaccaro Sep 17 '19 at 3:52
  • $\begingroup$ It's an $A$-action on $\Lambda$ (cf the Wikipedia page), not an $H$-action. I corrected accordingly. $\endgroup$ – YCor Sep 17 '19 at 6:09
  • $\begingroup$ This kind of action of a wreath product is often called "product action". In the finite setting, precise conditions for a subgroup of such a permutation group to be primitive are given as type III(b) in Liebeck–Praeger–Saxl. $\endgroup$ – Colin Reid Sep 17 '19 at 8:23
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Just a remark : if $A$ and $B$ are non trivial finite $p$-groups, each acting faithfully as transitive permutation groups, then the action of $ A \wr B $ is never primitive as a permutation action. For the transitive action of the wreath product is on at least $p^{2}$ points, so the point stabilizer in the action is not a maximal subgroup.

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Precise necessary and sufficient conditions for primitivity of the wreath product in product action are given in Lemma 2.7A of the book "Permutation Groups" of Dixon and Mortimer. Namely, the top group has to act transitively on a finite set, and the bottom group has to act primitively but not regularly.

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