finite subgroups of PGL(3,C)

The enumeration of finite subgroups of $\operatorname{PGL}_2(\mathbb{C})$ is one of the classic classification problems: mathematicians in many fields know well that the answer is cyclic groups, dihedral groups, $A_4$, $S_4$ and $A_5$. Moreover each of these groups occurs exactly once, up to conjugacy.

I hadn't thought about this classification very much for many years (if at all), but recently I have had some impetus to do so. In particular I am currently reading this wonderful note of A. Beauville, which treats the classification of finite subgroups of $\operatorname{PGL}_2(K)$ for any field $K$ of characteristic $0$. Here the new work is not so much figuring out exactly which groups can be realized over a given $K$, but rather working out the classification up to $K$-rational conjugacy, which turns out to be an interesting problem in Galois cohomology.

In some recent ruminations I have been thinking about Galois cohomology of finite subgroups of $\operatorname{PGL}_N(K)$, but it occurs to me that I don't know anything concrete past $N = 2$. Well, I can see that every finite group occurs as a subgroup of $\operatorname{PGL}_N(K)$ for any field $K$ and sufficiently large $N$, so obviously one can only ask for so much.

So...what about $\operatorname{PGL}_3(\mathbb{C})$? In particular:

1) What are the finite subgroups of $\operatorname{PGL}_3(\mathbb{C})$.

2) Is it still the case that if a finite group can be embedded in $\operatorname{PGL}_3(\mathbb{C})$, the embedding is unique up to conjugacy? If so, is there a general principle at work here?

Note that Beauville gives proofs of everything in the $\operatorname{PGL}_2$ case. But much of his classification arguments turn on the "accidental isomorphism" $\operatorname{PGL}_2 \cong SO(q)$, where $q = x^2 + yz$. Perhaps if these arguments generalized to $\operatorname{PGL}_N$ in some well-known way, he would have given proofs which generalize as well...

Solution 1:

As I recall, the finite subgroups of $PGL_N(\mathbb C)$ are classified for $N \le 7$. See Miller--Blichfeldt--Dickson's "Theory of finite groups" (1916) for $N=3$ or Blitchfeldt's "Finite collineation groups" (1917) for $N=3, 4$. (Beware: the terminology is quite old---for instance, isomorphic only means something like isomorphic modulo a normal (or maybe central) subgroup in modern notation.)

Feit (The current situation in the theory of finite simple groups. Actes du Congr`es International des Math'ematiciens, Nice 1970) gives a list of maximal finite subgroups in these cases, though there are a few cases that he missed in higher dimensions.

Incidentally, in my thesis, I tried classifying the finite solvable subgroups of $GSp_4(\mathbb C)$ in a more simple manner than Blitchfeldt, though at one crucial point, I couldn't get around using some of his work. Still, it might be useful (and easier to read than Blitchfeldt) if you want to know some basic techniques in these classification questions.

EDIT: It turns out I wrote down the list of primitive subgroups of $PGL_3(\mathbb C)$ in my thesis. (I knew I had it written down somewhere!)

http://thesis.library.caltech.edu/2141/1/Thesis.pdf

See chapter 8 for the list: it consists of 6 groups: three nonsimple of orders 36, 72 and 216; and $A_5 \simeq PSL(2,5)$, $A_6 \simeq PSL(2,9)$ and $PSL(2,7)$. See Appendix B for the GAP notation of the first 3 groups.

RE-EDIT: The paper

Ian Hambleton and Ronnie Lee, Finite group actions on $\mathbb P^2(\mathbb C)$, J. Alg. (1988)

has a description of all finite subgroups of $PGL_3(\mathbb C)$, including the 4 classes of imprimitive groups.

Solution 2:

On mathoverflow a similar question was asked. You can take the list for SU3 and mod out by the centers to get the list for PGL3. Several references for SU3 are given (in more than one answer), and some discussion of SU(n) is also given.

For instance A6 is in PGL3 because the Valentiner group 3.A6 is in SL3. I believe one of the papers referenced even goes so far as to list the groups by their image in PGL3. I remembered the discussion since that is where I learned the name of 3.A6, but I guess noone mentioned it at the time (so I am mentioning it here. VALENTINER!).