Editing Quotient group (section)

== Quotients of Lie groups ==
If <math>G</math> is a [[Lie group]] and <math>N</math> is a normal and closed (in the topological rather than the algebraic sense of the word) [[Lie subgroup]] of {{tmath|1= G }}, the quotient <math>G\,/\,N</math> is also a Lie group.  In this case, the original group ''<math>G</math>'' has the structure of a [[fiber bundle]] (specifically, a [[principal bundle|principal {{tmath|1= N }}-bundle]]), with base space <math>G\,/\,N</math> and fiber {{tmath|1= N }}. The dimension of <math>G\,/\,N</math> equals {{tmath|1= \dim G - \dim N }}.<ref>John M. Lee, Introduction to Smooth Manifolds, Second Edition, theorem 21.17</ref>

Note that the condition that <math>N</math> is closed is necessary. Indeed, if <math>N</math> is not closed then the quotient space is not a [[T1-space]] (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a [[Hausdorff space]].

For a non-normal Lie subgroup {{tmath|1= N }}, the space <math>G\,/\,N</math> of left cosets is not a group, but simply a [[differentiable manifold]] on which <math>G</math> acts.  The result is known as a [[homogeneous space]].