The finite subgroups of SO(3) are the dihedral groups and
the three symmetry groups of Platonic solids: the tetrahedron,
octahedron, and icosahedron groups, which are in turn the alternating
and symmetric groups A_{4}, S_{4}, and A_{5}.

The multiplication of quaternions leads nicely to a
determination of the finite subgroups of SO(4), the special orthogonal
group of dimension four real matrices. There are many infinite families
of such groups, and as for SO(3), some individual groups which relate to
symmetries of regular polytopes. We will derive this list and look at
some of the groups' properties and specific examples, including
symplectic groups and symmetry groups of regular four polytopes and
quotients by these groups.

This is a background talk for talks I'll give early next
semester on orbifolds. There I'll focus on low dimensional orbifolds,
particularly those of dimensions four and five, vertex group homology in
those degrees, and orbifold cobordism and invariants.

This talk is meant to be accessible to all graduate students.