Editing Orthogonal group (section)

=== Discrete subgroups ===
As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.<ref group="note">Infinite subsets of a compact space have an [[accumulation point]] and are not discrete.</ref> These subgroups are known as [[point group]]s and can be realized as the symmetry groups of [[polytope]]s. A very important class of examples are the [[finite Coxeter group]]s, which include the symmetry groups of [[regular polytope]]s.

Dimension 3 is particularly studied – see [[point groups in three dimensions]], [[polyhedral group]]s, 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]] {{math|A<sub>''n''</sub>}})
* [[Signed permutation matrices]] (the [[Coxeter group]] {{math|B<sub>''n''</sub>}}); also equals the intersection of the orthogonal group with the [[integer matrix|integer matrices]].<ref group="note">{{math|O(''n'') ∩ [[general linear group|GL]](''n'', '''Z''')}} equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be {{math|±1}} (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.</ref>