Finite subgroups of SU(2)

This project aims at the classification of the finite subgroups of the group SU(2) of 2x2 unitary matrices with unit determinant.

In the first part of the project you will classify the finite rotation groups in three dimensions; that is, the finite subgroups of the group SO(3) of 3x3 orthogonal matrices of unit determinant.

This can be found, for example, in any of the first three textbooks listed in the references: pages 35-37 in Rees, chapter 19 in Armstrong or chapter 2 in Grove and Benson.

In the second part of the project you will classify the finite subgroups of SU(2) by exploiting the (two-to-one) homomorphism SU(2) → SO(3). This can in turn be realised very concretely using quaternions, as described in chapter 6 of Coxeter.

The classification can be neatly summarised in terms of the Dynkin diagrams of the ADE series via the McKay correspondence.

References

1. Notes on Geometry
   Elmer Rees
2. Groups and Symmetries
   Mark Armstrong
3. Finite reflection groups
   Larry Grove and Clark Benson
4. Regular complex polytopes
   Harold Coxeter