Editing Lie group (section)

=== Lie subgroup ===
A '''Lie subgroup''' <math>H</math> of a Lie group <math>G</math> is a Lie group that is a [[subset]] of <math>G</math> and such that the [[inclusion map]] from <math>H</math> to <math>G</math> is an [[injective]] [[Immersion (mathematics)|immersion]] and [[group homomorphism]]. According to [[Closed subgroup theorem|Cartan's theorem]], a closed [[subgroup]] of <math>G</math> admits a unique smooth structure which makes it an [[embedding|embedded]] Lie subgroup of <math>G</math>—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.

Examples of non-closed subgroups are plentiful; for example take <math>G</math> to be a torus of dimension 2 or greater, and let <math>H</math> be a [[one-parameter subgroup]] of ''irrational slope'', i.e. one that winds around in ''G''. Then there is a Lie group [[homomorphism]] <math>\varphi:\mathbb{R}\to G</math> with {{tmath|1= \mathrm{im}(\varphi) = H }}. The [[closure (topology)|closure]] of <math>H</math> will be a sub-torus in {{tmath|1= G }}.

The [[exponential map (Lie theory)|exponential map]] gives a [[Lie group–Lie algebra correspondence#The correspondence|one-to-one correspondence]] between the connected Lie subgroups of a connected Lie group <math>G</math> and the subalgebras of the Lie algebra of {{tmath|1= G }}.<ref>{{harvnb|Hall|2015}} Theorem 5.20</ref>  Typically, the subgroup corresponding to a subalgebra is not a  closed subgroup.  There is no criterion solely based on the structure of <math>G</math> which determines which subalgebras correspond to closed subgroups.