The finite subgroups of SU(n)

There is an algorithm due to Zassenhaus which, in principle, lists all conjugacy classes of finite subgroups of compact Lie groups. I believe that the algorithm was used for $\mathrm{SO}(n)$ for at least $n=6$ if not higher. I believe it is expensive to run, which means that in practice it is only useful for low dimension.


Added

Now that I'm in my office I have my orbifold folder with me and I can list some relevant links:

  1. Zassenhaus's original paper (in German) Über einen Algorithmus zur Bestimmung der Raumgruppen
  2. There is a book by RLE Schwarzenberger N-dimensional crystallography with lots of references
  3. There are a couple of papers in Acta Cryst. by Neubüser, Wondratschek and Bülow titled On crystallography in higher dimensions
  4. There is a sequence of papers in Math. Comp. by Plesken and Pohst titled On maximal finite irreducible subgroups of GL(n,Z) which I remember were relevant.

Independent of this algorithm, there is some work on $\mathrm{SU}(n)$ from the physics community motivated by elementary particle physics and more modern considerations of the use of orbifolds in the gauge/gravity correspondence.

The case of $\mathrm{SU}(3)$ was done in the mid 1960s and is contained in the paper Finite and Disconnected Subgroups of SU(3) and their Application to the Elementary-Particle Spectrum by Fairbairn, Fulton and Klink. For the case of $\mathrm{SU}(4)$ there is a more recent paper A Monograph on the Classification of the Discrete Subgroups of SU(4) by Hanany and He, and references therein.


Further edit

The paper Non-abelian finite gauge theories by Hanany and He have the correct list of finite subgroups of SU(3), based on Yau and Yu's paper Gorenstein quotient singularities in dimension three.


The finite subgroups of SU(3) have been known for a century. I think you can find it in these references (my Departmental library does not go back this far):

MR1500676 Blichfeldt, H. F. On the order of linear homogeneous groups. II. Trans. Amer. Math. Soc. 5 (1904), no. 3, 310--325. (doi:10.1090/S0002-9947-1904-1500676-6)

MR1511301 Blichfeldt, H. F. The finite, discontinuous primitive groups of collineations in four variables. Math. Ann. 60 (1905), no. 2, 204--231. (EuDML)

MR1560123 Blichfeldt, H. F. Blichfeldt's finite collineation groups. Bull. Amer. Math. Soc. 24 (1918), no. 10, 484--487. (Project Euclid, open access)

and also in this book

MR0123600 (23 #A925) Miller, G. A. ; Blichfeldt, H. F. ; Dickson, L. E. Theory and applications of finite groups. Dover Publications, Inc., New York 1961 xvii+390 pp.


This is really a comment on the top answer above, but since new users can't comment, I'll let someone else manually transfer the information to the right place.

There is a further mistake in the list of Fairbairn, Fulton and Klink (repeated in the list of Hanahy and He), which appears to be a misunderstanding of the classification by Blichfeldt et al. Two of the cases in that classification consists of semidirect products of abelian groups by $A_3$ and $S_3$. However, it is not specified which abelian groups can occur in this fashion!

Fairbairn, Fulton and Klink mistakenly assume that the abelian group in question has to be $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n \mathbb{Z}$ for some positive integer $n$, thus giving rise to the groups they denote $\Delta(3n^2)$ and $\Delta(6n^2)$. However, this is not the case.

Example 1: $A_3$ acts on the copy of $\mathbb{Z}/7\mathbb{Z}$ generated by the diagonal matrix with entries $e^{2\pi i/7}, e^{4 \pi i/7}, e^{8 \pi i/7}$; this example occurs inside the exceptional subgroup of order 168. More generally, if $m,n$ are positive integers and $m^2+m+1 \equiv 0 \pmod{n}$, then $A_3$ acts on the copy of $\mathbb{Z}/n\mathbb{Z}$ generated by the diagonal matrix with entries $e^{2\pi i/n}, e^{2m \pi i/n}, e^{2m^2 \pi i/n}$.

Example 2: $S_3$ acts on the copy of $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z}$ generated by the diagonal matrices with entries $e^{2\pi i/9}, e^{2\pi i/9}, e^{14 \pi i/9}$ and $1, e^{2\pi i/3}, e^{4\pi i/3}$; this example occurs inside the exceptional subgroup of order 648.

I don't know a reference for the complete classification of the abelian groups that can occur inside the semidirect product. Yau and Yu don't say any more than Blichfeldt et al, though they do at least provide a helpful rewrite of the classification in modern language.