This post is a dual version for the [Generalization of a theorem of Øystein Ore](https://mathoverflow.net/q/179555/34538) in which [it's proved](https://mathoverflow.net/a/195331/34538):  

*Theorem*: Let $[H, G]$ be a distributive interval of finite groups. Then $\exists g \in G$ such that $\langle H,g \rangle = G$. 


*Definition*: Let $W$ be a representation of $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.  

 Let the *fixed-point subspace* $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.  

Let the *pointwise stabilizer subgroup* $G\_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$. 


*Definition*: $[H,G]$ is called linearly primitive if $\exists V$ irred. complex repr. of $G$ with $G\_{(V^H)} = H$. 
 *Remark:* $[1,G]$ is linearly primitive iff $G$ is [linearly primitive](http://groupprops.subwiki.org/wiki/Linearly_primitive_group).


**Question**: Let $[H, G]$ be a distributive interval of finite groups. Is $[H,G]$ linearly primitive? 


*Remark:* The case $H = 1$ is true because $[1,G]=\mathcal{L}(G)$ is distributive iff $G$ is cyclic, but an abelian group is linearly primitive iff it is cyclic. It follows that it's also true if $H \triangleleft G$. 


It's true by GAP for $|G:H|<32$ or $\vert G \vert \le 1000$ or $G$ perfect with $\vert G \vert < 10080$ (except $7680$). 



A dual version of a theorem of Øystein Ore in group theory
**Yes.** 


This was proved in the [planar algebra](https://en.wikipedia.org/wiki/Planar_algebra) framework, see [arXiv:1704.00745](https://arxiv.org/pdf/1704.00745v3.pdf), Corollary 6.10. 


For a self-contained group-theoretic proof, see [arXiv:1708.02565](https://arxiv.org/abs/1708.02565).


