Editing Orthogonal group (section)

== Related groups ==
The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.

The inclusions {{math|O(''n'') ⊂ U(''n'') ⊂ USp(2''n'')}} and {{math|USp(''n'') ⊂ U(''n'') ⊂ O(2''n'')}} are part of a sequence of 8 inclusions used in a [[Bott periodicity theorem#Geometric model of loop spaces|geometric proof of the Bott periodicity theorem]], and the corresponding quotient spaces are [[symmetric space]]s of independent interest – for example, {{math|U(''n'')/O(''n'')}} is the [[Lagrangian Grassmannian]].

=== Lie subgroups ===
In physics, particularly in the areas of [[Kaluza–Klein]] compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:
: <math>\mathrm{O}(n) \supset \mathrm{O}(n - 1)</math> – preserve an axis
: <math>\mathrm{O}(2n) \supset \mathrm{U}(n) \supset \mathrm{SU}(n)</math> – {{math|U(''n'')}} are those that preserve a compatible complex structure ''or'' a compatible symplectic structure – see [[Unitary group#2-out-of-3 property|2-out-of-3 property]]; {{math|SU(''n'')}} also preserves a complex orientation.
: <math>\mathrm{O}(2n) \supset \mathrm{USp}(n)</math>
: <math>\mathrm{O}(7) \supset \mathrm{G}_2</math>

=== Lie supergroups ===
The orthogonal group {{math|O(''n'')}} is also an important subgroup of various Lie groups:
: <math>\begin{align}
     \mathrm{U}(n)    &\supset \mathrm{O}(n) \\
     \mathrm{USp}(2n) &\supset \mathrm{O}(n) \\
     \mathrm{G}_2 &\supset \mathrm{O}(3) \\
     \mathrm{F}_4 &\supset \mathrm{O}(9) \\
     \mathrm{E}_6 &\supset \mathrm{O}(10) \\
     \mathrm{E}_7 &\supset \mathrm{O}(12) \\
     \mathrm{E}_8 &\supset \mathrm{O}(16)
\end{align}</math>

==== Conformal group ====
{{Main|Conformal group}}

Being [[isometry|isometries]], real orthogonal transforms preserve [[angle]]s, and are thus [[conformal map]]s, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between [[congruence (geometry)|congruence]] and [[similarity (geometry)|similarity]], as exemplified by SSS (side-side-side) [[Congruence (geometry)#Congruence of triangles|congruence of triangles]] and AAA (angle-angle-angle) [[similar triangles|similarity of triangles]]. The group of conformal linear maps of {{math|'''R'''<sup>''n''</sup>}} is denoted {{math|CO(''n'')}} for the '''conformal orthogonal group''', and consists of the product of the orthogonal group with the group of [[Homothetic transformation|dilations]]. If {{math|''n''}} is odd, these two subgroups do not intersect, and they are a [[direct product of groups|direct product]]: {{math|1=CO(2''k'' + 1) = O(2''k'' + 1) × '''R'''<sup>∗</sup>}}, where {{math|1='''R'''<sup>∗</sup> = '''R'''∖{0}}} is the real [[multiplicative group]], while if {{math|''n''}} is even, these subgroups intersect in {{math|±1}}, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: {{math|1=CO(2''k'') = O(2''k'') × '''R'''<sup>+</sup>}}.

Similarly one can define {{math|CSO(''n'')}}; this is always: {{math|1=CSO(''n'') = CO(''n'') ∩ GL<sup>+</sup>(''n'') = SO(''n'') × '''R'''<sup>+</sup>}}.

=== 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>

=== Covering and quotient groups ===
The orthogonal group is neither [[simply connected]] nor [[centerless]], and thus has both a [[covering group]] and a [[quotient group]], respectively:
* Two covering [[Pin group]]s, {{math|Pin<sub>+</sub>(''n'') → O(''n'')}} and {{math|Pin<sub>−</sub>(''n'') → O(''n'')}},
* The quotient [[projective orthogonal group]], {{math|O(''n'') → PO(''n'')}}.
These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:
* [[Spin group]], {{math|Spin(''n'') → SO(''n'')}},
* [[Projective special orthogonal group]], {{math|SO(''n'') → PSO(''n'')}}.
Spin is a 2-to-1 cover, while in even dimension, {{math|PSO(2''k'')}} is a 2-to-1 cover, and in odd dimension {{math|PSO(2''k'' + 1)}} is a 1-to-1 cover; i.e., isomorphic to {{math|SO(2''k'' + 1)}}. These groups, {{math|Spin(''n'')}}, {{math|SO(''n'')}}, and {{math|PSO(''n'')}} are Lie group forms of the compact [[special orthogonal Lie algebra]], <math>\mathfrak{so}(n, \mathbf{R})</math> – {{math|Spin}} is the simply connected form, while {{math|PSO}} is the centerless form, and {{math|SO}} is in general neither.<ref group="note">In odd dimension, {{math|SO(2''k'' + 1) ≅ PSO(2''k'' + 1)}} is centerless (but not simply connected), while in even dimension {{math|SO(2''k'')}} is neither centerless nor simply connected.</ref>

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.