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Is there a classification of subgroups $G$ of $\operatorname{GL}(n,q)$ which act transitively on $\mathbb{F}_q^n \setminus \{0\}$, the set of non-zero vectors?

Any $G$ with $\operatorname{GL}(n/m,q^m) \leq G \leq \Gamma \operatorname{L}(n/m,q^m)$ for some $m \mid n$ is an example, since then $G$ contains a Singer cycle. Another example is $\operatorname{Sp}(2d,q)$ in $\operatorname{GL}(2d,q)$.

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    $\begingroup$ There is a list of transitive finite linear groups in Wikipedia. $\endgroup$
    – Friedrich Knop
    Commented Sep 11, 2021 at 9:59
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    $\begingroup$ These is essentially equivalent to the classification of finite double transitive groups of affine type. There is a long discussion of this question here. The consensus appears to be that they were classified by Hering, but there appears to be a dearth of precise references. $\endgroup$
    – Derek Holt
    Commented Sep 11, 2021 at 10:19
  • $\begingroup$ In Thm 15.1 (p. 197) of iazd.uni-hannover.de/fileadmin/iazd/sambale/pdfs/lnm.pdf there is a precise list of the exceptional groups including ids to libraries. $\endgroup$
    – Brauer Suzuki
    Commented Sep 11, 2021 at 19:38
  • $\begingroup$ Martin Liebeck's paper "The affine permutation groups of rank three" contains an appendix where he states and proves Hering's theorem. This theorem gives an explicit list of the groups you are interested in. I have an e-copy of the paper and can email it to you if you want... $\endgroup$
    – Nick Gill
    Commented Sep 13, 2021 at 8:17

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