Recall first the following result of Iwasawa and its corollaries. In what follows, $G$ is a group acting primitively on a set $E$, and $x_0\in E$. We also assume that there exists a normal subgroup $A$ of $Stab_G(x_0)$ such that the various conjugates of $A$ generate $G$.
Thm.(Iwasawa) Under the previous assumptions, for any normal subgroup $N$ of $G$, either $N$ acts trivially on $E$, or $G/N$ is isomorphic to a quotient of $A$.
Corollary 1. If $G=[G,G]$ and $A$ is solvable, then all proper normal subgroups act trivially on $E$, or $G$ is simple.
Corollary 2. If $G=[G,G]$ and $A$ is solvable, and if $H$ denotes the kernel of the action of $G$ on $E$, then $G/H$ is simple.
Corollary $2$ is a powerful criterion to show that a group $G$ is simple, but the only application to simplicity I know is the case of $PSL(V)$, where $V$ is a finite dimensional $K$-vector space of dimension $n$ ($K$ is any field) is simple (except for $n=2$ and $\vert K\vert\leq 3$) using Corollary 2.
After giving it a bit of thinking, I convinced myself that the simplicity of the alternating group $A_n$ for $n\geq 6$ may be proven using Corollary 2 by letting $A_n$ acts on the set of transpositions, even if I didn't find any reference using this approach.
And that's it...Of course, I'm not a group theorist, so it's more likely that I don't know other classical applications of Iwasawa criterion, and I really would like to learn them. Hence the following:
Question. Which other group may be proven to be simple using the results above ? I'm not assuming $G$ to be finite.
For example, can we prove that $SO_{2n+1}(\mathbb{R})$ and $PSO_{2n}(\mathbb{R})$ are simple using Corollary 2 (or 1, or the general thm) ?
The only proof I know for $SO$ reduces to the case $n=3$ (like simplicity of $A_n$ can be reduced to the simplicity of $A_5$)
Same question for $PSp_{2n}(K)$, where $K$ is an arbitrary field (except for the few exceptional cases). I suspect we could use symplectic transvections, and we can make this group to act on lines, but i didn't check.
Once again, the only proof i know reduces to the case of $PSp_2(K)$.
I'm also interested by any other example which is not cited above (references are welcome !)
