Closed-subgroup theorem
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In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding.[1][2][3] One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan,[4] who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.[5][6]