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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 $\operatorname{GL}(2d,q)$.

Carl-Fredrik Nyberg Brodda
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    There is a list of transitive finite linear groups in Wikipedia. – Friedrich Knop Sep 11 '21 at 09:59
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    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. – Derek Holt Sep 11 '21 at 10:19
  • In Thm 15.1 (p. 197) of https://www.iazd.uni-hannover.de/fileadmin/iazd/sambale/pdfs/lnm.pdf there is a precise list of the exceptional groups including ids to libraries. – Brauer Suzuki Sep 11 '21 at 19:38
  • 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... – Nick Gill Sep 13 '21 at 08:17

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