Editing Homogeneous space (section)

== Homogeneous spaces as coset spaces ==
In general, if ''X'' is a homogeneous space of ''G'', and ''H''<sub>''o''</sub> is the [[stabilizer (group theory)|stabilizer]] of some marked point ''o'' in ''X'' (a choice of [[origin (mathematics)|origin]]), the points of ''X'' correspond to the left [[coset]]s ''G''/''H''<sub>''o''</sub>, and the marked point ''o'' corresponds to the coset of the identity. Conversely, given a coset space ''G''/''H'', it is a homogeneous space for ''G'' with a distinguished point, namely the coset of the identity. Thus a homogeneous space can be thought of as a coset space without a choice of origin.

For example, if ''H'' is the identity subgroup {{mset|''e''}}, then ''X'' is the [[principal homogeneous space|''G''-torsor]], which explains why ''G''-torsors are often described intuitively as "''G'' with forgotten identity".

In general, a different choice of origin ''o'' will lead to a quotient of&nbsp;''G'' by a different subgroup ''H<sub>o′</sub>'' that is related to ''H<sub>o</sub>'' by an [[inner automorphism]] of&nbsp;''G''.  Specifically,
{{NumBlk||<math display="block">H_{o'} = gH_og^{-1}</math>|{{EquationRef|1}}}}
where ''g'' is any element of ''G'' for which {{nowrap|1=''go'' = ''o''′}}. Note that the inner automorphism (1) does not depend on which such ''g'' is selected; it depends only on ''g'' modulo&nbsp;''H''<sub>''o''</sub>.

If the action of ''G'' on ''X'' is [[continuous map|continuous]] and ''X'' is [[Hausdorff space|Hausdorff]], then ''H'' is a [[closed subgroup]] of ''G''. In particular, if ''G'' is a [[Lie group]], then ''H'' is a [[Lie subgroup]] by [[Closed subgroup theorem|Cartan's theorem]].  Hence {{nowrap|''G'' / ''H''}} is a [[smooth manifold]] and so ''X'' carries a unique [[smooth structure]] compatible with the group action.

One can go further to [[double coset|''double'' coset]] spaces, notably [[Clifford–Klein form]]s Γ\''G''/''H'', where Γ is a discrete subgroup (of ''G'') acting [[properly discontinuously]].