Spin Group - Discrete Subgroups

Discrete Subgroups

Discrete subgroups of the spin group can be understood by relating them to discrete subgroups of the special orthogonal group (rotational point groups).

Given the double cover by the lattice theorem, there is a Galois connection between subgroups of Spin(n) and subgroups of SO(n) (rotational point groups): the image of a subgroup of Spin(n) is a rotational point group, and the preimage of a point group is a subgroup of Spin(n), and the closure operator on subgroups of Spin(n) is multiplication by {±1}. These may be called "binary point groups"; most familiar is the 3-dimensional case, known as binary polyhedral groups.

Concretely, every binary point group is either the preimage of a point group (hence denoted for the point group ), or is an index 2 subgroup of the preimage of a point group which maps (isomorphically) onto the point group; in the latter case the full binary group is abstractly (since is central). As an example of these latter, given a cyclic group of odd order in SO(n), its preimage is a cyclic group of twice the order, and the subgroup maps isomorphically to

Of particular note are two series:

• higher binary tetrahedral groups, corresponding to the 2-fold cover of symmetries of the n-simplex.

  This group can also be considered as the double cover of the symmetric group, with the alternating group being the (rotational) symmetry group of the n-simplex.

• higher binary octahedral groups, corresponding to the 2-fold covers of the hyperoctahedral group (symmetries of the hypercube, or equivalently of its dual, the cross-polytope).

For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group.