Smallest group acting transitively on projective space

Question

Let $ K $ be a field. Let $ K^n $ be an $ n $ dimensional vector space over $ K $. Let $ KP^{n-1} $ be the projective space of lines in $ K^n $. Let $ GL(n,K) $ be the group of invertible $ n \times n $ matrices over $ K $. What is the smallest subgroup of $ GL(n,K) $ that acts transitively on $ KP^{n-1} $?

Context: When $ K=\mathbb{C} $ then I think the smallest subgroup of $ GL(n,\mathbb{C}) $ acting transitively on $ \mathbb{C}P^{n-1} $ is $ SU(n) $. When $ K=\mathbb{R} $ then I think the smallest subgroup of $ GL(n,\mathbb{R}) $ acting transitively on $ \mathbb{R}P^{n-1} $ is $ SO(n) $. I was wondering if this is true and also what the corresponding group is for other choices of $ K $.

Answer 1

For $K = \Bbb R$ it follows from Montgomery & Samelson, "Transformation Groups of Spheres" that one can do better than $\operatorname{SO}(n)$ when $n \equiv 0 \pmod 2$ (and $n > 2$) or $n = 7$.

• If $n \equiv 0 \pmod 2$ and $n > 2$, $\operatorname{SU}\left(\frac n2\right)$, which has dimension $\frac{1}{4} n^2 - 1$, acts transitively on $\Bbb R P^{n - 1}$.

• If $n \equiv 0 \pmod 4$, $\operatorname{Sp}\left(\frac n4\right)$, which has dimension $\frac18 n (n + 2)$, acts transitively on $\Bbb R P^{n - 1}$.

  • In the special case $n = 16$, $\operatorname{Spin}(9)$, which has the same dimension ($36$) as $\operatorname{Sp}(4)$, also acts transitively on $\Bbb R P^{15}$.
  • In the special case $n = 8$, $\operatorname{Spin}(7)$, which has dimension $21$, acts transitively on $\Bbb R P^7$; it is smaller than $\operatorname{SO}(8)$ (dimension $28$) but it's still larger than $\operatorname{Sp}(2)$ (dimension $10$).

• If $n = 7$, $\operatorname{G}_2$, which has dimension $14$, acts transitively on $\Bbb R P^6$.

Montgomery, Deane, and Hans Samelson. "Transformation Groups of Spheres." Annals of Mathematics, vol. 44, no. 3, 1943, pp. 454–70. JSTOR, https://doi.org/10.2307/1968975.

Answer 2

Over finite fields you get much more. For example (images of) the Singer cycle (and overgroups). As a somewhat random example, there are (up to conjugacy) 7 minimal transitive subgroups of $PGL(4,3)$, one of which is the image of a Singer cycle of order $40$ and the others up to order $360$.