Editing Orthogonal group (section)

=== 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>}}.