What are the discrete subgroups of the quaternions?

Question

I'm trying to find all the discrete subgroups of the quaternions under multiplication. Particularly I'm interested in subgroups of unit quaternions (if we have a non-unit quaternion then we have terms accumulating near $0$, so it's not discrete. These groups are still interesting, but I want to ignore them for now.)

First we have some trivial groups, by which I mean they are contained in a plane:

$$\{\cos(2\pi k/n) + v \sin(2\pi k/n)\},$$

where $v$ is any unit quaternion with no real part. These groups are isomorphic to the complex $n$th roots of unity.

As for non trivial discrete groups, I have two so far: $$\{\pm 1, \pm i, \pm j, \pm k\}$$ with 8 elements, and the group generated by $$\{(1+i)/\sqrt{2}, (1+j)/\sqrt{2}\}$$ with $48$ elements.

Are there any more?

Answer 1

Due to some interesting geometric coincidences, you can visualize some special discrete subgroups of the unit quaternions in the following manner.

First, the unit quaternions $\{a + bi + (c+di)j \mid a^2 + b^2 + c^3 + d^2 = 1\}$ form the unit sphere $S^3$ in $\mathbb C^2 = \mathbb R^4$.

Second, the group of rigid motions of the unit sphere $S^2$ in $\mathbb R^3$ is isomorphic to the special orthogonal group $SO(3)$, consisting of all orthonormal $3 \times 3$ matrices of real numbers with determinant $+1$.

Third, the fundamental group of $SO(3)$ is cyclic of order 2 (which is a story in itself).

Fourth, there is a map $S^3 \mapsto SO(3)$ which (to a topologist) is a 2--1 universal covering map and (to a group theorist) is a 2--1 surjective homomorphism using unit quaternion group structure on its domain. The kernel of this homomorphism is $\pm 1$.

Now we have all the pieces: Given any finite subgroup of the orientation preserving rigid motions of $S^2$, its pre-image under the map $S^3 \mapsto SO(3)$ is a finite subgroup of the unit quaternions, having twice the order of the given subgroup.

With this in hand, here are a few interesting finite subgroups of rigid motions of $S^2$:

• The orientation preserving symmetries of a tetrahedron, a group of order 12, whose pre-image is an order 24 subgroup of unit quaternions.
• The orientation preserving symmetries of a cube (or an octahedron), a group of order 24, whose preimage is an order 48 subgroup of the unit quaternions (which I believe is the one you described in your question; and to be honest, this one contains the previous one with index $2$, as one can see by a standard embedding of the tetrahedron in the cube).
• The orientation preserving symmetries of a dodecahedron (or an icosahedron), a group of order $60$, whose pre-image is an order $120$ subgroup of the unit quaternions.

Answer 2

By considering the projective image to $\mathbb H^\times/\mathbb R^\times \simeq SO(3)$, it suffices to look at discrete subgroups of SO(3)---the discrete subgroups of $\mathbb H^\times$ will just be "discrete pullbacks" of these groups.

By compactness of SO(3), the only discrete subgroups you get are the finite ones. These have be classified classically (probably going back to Klein): cyclic, dihedral, $A_4$, $S_4$ and $A_5$---see Lee's answer for more details.

If you allow yourself to work with indefinite quaternion algebras (so not the Hamilton quaternions), you can get much more interesting discrete subgroups: the arithmetic Fuchsian groups. A good reference for the latter is the end of Svetlana Katok's book on Fuchsian groups.