Assume everything is finite.


Let $G$ be a primitive permutation group with point stabiliser $G\_\alpha$ for some $\alpha$. For $\beta\ne\alpha$, by an arc stabiliser we mean $G\_{\alpha\beta}=G\_\alpha\cap G\_\beta$ and an edge stabiliser we mean $G\_{\{\alpha,\beta\}}$, the stabiliser of the set $\{\alpha,\beta\}$. Note that $G\_{\{\alpha,\beta\}}\ge G\_{\alpha\beta}$.


**My question is:** can an arc stabiliser $G\_{\alpha\beta}\ne 1$ be normal in $G\_\alpha$?


This is impossible if $G\_{\{\alpha,\beta\}}> G\_{\alpha\beta}$. In this case, there exists $g\in G$ such that $\alpha^g=\beta$ and $\beta^g=\alpha$. It follows that $g\in N\_G(G\_{\alpha\beta})$. Note that $G\_\alpha$ is maximal in $G$ and $G\_{\alpha\beta}$ cannot be normal in $G$ (otherwise $G\_{\alpha\beta}$ will be in the kernel of this action), the normaliser $N\_G(G\_{\alpha\beta})=G\_\alpha$. This gives a contradiction: $g\in G\_\alpha$ but $\alpha^g=\beta\ne \alpha$.


I think there might exist an example in the case when $G\_{\{\alpha,\beta\}}=G\_{\alpha\beta}$. However, by a quick check with Magma there is no such primitive permutation group $G$ with $|G|\le 300$.




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An equivalent statement is: Let $G$ be a group and $H$ a maximal and core-free subgroup of $G$. Is it possible that $1\ne H\cap H^g\lhd H$ for some $g\notin H$?


As shown in the comments there by Verret and Holt there is no such example for degree $\le 4095$. I also think the proof or an example, if any, will not be elementary (for example, applying O'Nan-Scott Theorem).




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This was initially a [MSE question](https://math.stackexchange.com/questions/3835339/) I asked.



Can $1\ne H\cap H^g\lhd H$ happen if $G$ is a primitive permutation group with stabiliser $H$?
An example was constructed by Pablo Spiga:
<https://arxiv.org/abs/2102.13614>
"A generalization of Sims conjecture for finite primitive groups and two point stabilizers in primitive groups"


