Closed-subgroup theorem
= Closed-subgroup theorem
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= Cartan closed-subgroup theorem
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Every closed <subgroup> of a finite-dimensional <Lie group> is an <embedded submanifold> and a <Lie subgroup>. Closedness alone forces the compatible smooth structure; smoothness need not be assumed in advance. The proof constructs the <Lie algebra of a closed subgroup> from <one-parameter subgroups>, then uses <local product coordinates for a closed subgroup>.