# Smallest group acting transitively on projective space

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 $$K$$.

• In the second case you probably mean "$K=\mathbb R$"? Feb 20 at 23:17
• @red_trumpet yes fixed! Feb 20 at 23:18
• If $n$ is even, then $\operatorname{SU}\left(\frac n2\right) < \operatorname{SO}(n)$ acts transitively on $\Bbb R P^{n - 1}$, and if $n \equiv 0 \pmod 4$, then $\operatorname{Sp}\left(\frac n4\right) < \operatorname{SO}(n)$ acts transitively on $\Bbb R P^{n - 1}$. For $n = 7$, $\operatorname{G}_2 < \operatorname{SO}(7)$ (dimension $14$) acts transitively on $\Bbb R P^6$. See Borel's classification of Lie groups acting transitively on spheres. Feb 20 at 23:37
• Oh that's very interesting so $S^7$ has a transitive action by $SO(8)$ of dim $28$ and $SU(4)$ of dim $15$ and $Sp(2)$ of dim $10$ and is even an H space in a natural way by identifying it with the unit octonions which have dimension $7$ (although they are not a group because they lack associativity) Feb 20 at 23:54
• Yes, and $S^7$ also admits a transitive action by $\operatorname{Spin}(7)$ (dimension $21$); that phenomenon is special to that dimension for reasons connected to triality. en.wikipedia.org/wiki/Triality Feb 21 at 1:47

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 $$360$$.

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.

• Maybe also worth mentioning are the nonminimal groups $\operatorname{Sp}\left(\frac n4\right) \cdot \operatorname{Sp}(1)$, of dimension $\frac18 n (n + 2) + 3$, which respectively act transitively on $\Bbb R P^{n - 1}$, $n \equiv 0 \pmod 4$. Feb 21 at 2:03
• Interestingly, the sequence $a_n := \operatorname{min}\{\dim G \mid \textrm{$G$acts transitively on$\Bbb S^{n - 1}$}\}$, $0,1,3,3,10,8,14,10,36,24,\ldots$, does not appear in the OEIS. Feb 21 at 16:29