Closed-subgroup theorem

Every closed subgroup of a Lie group is an embedded Lie subgroup, with its own smooth manifold structure inherited compatibly from the ambient group. It is also known as Cartan’s theorem.

Connect

Lie subgroup: A subgroup of a Lie group that is itself a Lie group under the induced operations. This is the structure the theorem guarantees for every closed subgroup.
Topological group: A group with a topology for which multiplication and inversion are continuous. Lie groups add smooth structure to this broader setting.
Élie Cartan: A French mathematician whose work shaped Lie theory, differential geometry, and the theory of continuous groups. The closed-subgroup theorem is also called Cartan’s theorem.
Stabilizer subgroup: The subgroup of transformations that fixes a chosen point or object. Stabilizers of smooth group actions are closed and therefore Lie subgroups.
Dense subgroup: A subgroup whose closure is the entire ambient topological group. A dense subgroup need not be closed, so the theorem gives it no embeddedness guarantee.
Exponential map: A map from a Lie algebra to its Lie group that sends infinitesimal generators to one-parameter subgroups. Local coordinates from this map help reveal smooth structure near the identity.
Closed set: A subset whose complement is open, equivalently one containing all its limit points. The subgroup hypothesis is topological closure in the ambient Lie group.
Sophus Lie: A Norwegian mathematician who founded the theory of continuous transformation groups and Lie groups. The theorem resolves a structural question in the theory Lie initiated.
Homogeneous space: A space on which a group acts transitively, often represented as a quotient by a stabilizer. Closed stabilizers provide Lie subgroups for constructing smooth quotient spaces.
Irrational line on a torus: The image of an irrational-slope one-parameter subgroup in a two-dimensional torus, dense but not closed. It exemplifies an immersed subgroup whose image fails to be embedded.