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This seems like it should be known, but I couldn't find a reference.

Let $V$ be a finite vector space and let $G$ be a group of semilinear maps on $V$ (i.e. linear composed with a field automorphism). Then $G$ acts naturally by semilinear maps on the dual space $V^*$ of linear maps from $V$ to $\mathbb{F}$. Suppose $G$ acts transitively on $V \setminus \{0\}$. Then certainly $G$ is transitive on codimension $1$ subspaces of $V^*$ (each is the annihilator of a nonzero element of $V$).

In which cases is $G$ transitive on $V^* \setminus \{0\}$?

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    $\begingroup$ Is this result known to be true for groups of linear maps of $V$? $\endgroup$
    – Derek Holt
    Commented Dec 10, 2022 at 9:39
  • $\begingroup$ I don't know; for the intended application I think I need to understand semilinear actions. If we're just given V as an elementary abelian group, there will be multiple choices of dual based on the choice of field. But in any case, I am imagining there would be some case by case description based on the classification of finite 2-transitive affine groups. $\endgroup$
    – Colin Reid
    Commented Dec 10, 2022 at 10:11

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