# Which groups can be proven to be simple using Iwasawa criterion?

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 !)

• My memory is that the simplicity proofs of all of the finite classical simple groups in Huppert's book "Endliche Gruppen I" use Iwasawa's lemma. I can check on Monday. Commented Sep 21, 2019 at 12:06
• The case of the groups ${\rm PSL}_n(F)$ for $n \geq 2$ and $|F| > 3$ is in kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf. Larry Grove's "Classical Groups and Geometric Algebra" proves simplicity of lots of classical groups using Iwasawa's criterion. He gives the criterion on page 4 and says it "will be applied many times throughout the book".
– KCd
Commented Feb 16, 2021 at 10:48
• This is exactly what I was looking for. Thanks a lot !!! Commented Feb 16, 2021 at 11:58