Cartan's theorem may refer to several mathematical results by Élie Cartan:
Closed-subgroup theorem, 1930, that any closed subgroup of a Lie group is a Lie subgroup Theorem of the highest weight, that the irreducible representations of Lie algebras or Lie groups are classified by their highest weights Lie's third theorem, an equivalence between Lie algebras and simply-connected Lie groups
See also Cartan's theorems A and B, c.1931 results by Henri Cartan concerning a coherent sheaf on a Stein manifold Cartan's lemma, several results by Élie or Henri Cartan Cartan–Dieudonné theorem, a result on orthogonal transformations and reflections