Discrete Subgroups
As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. These subgroups are known as point group and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.
Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.
Other finite subgroups include:
- Permutation matrices (the Coxeter group An)
- Signed permutation matrices (the Coxeter group Bn); also equals the intersection of the orthogonal group with the integer matrices.
Read more about this topic: Orthogonal Group, Related Groups
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