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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A locally closed set with a smooth group structure is a Lie group


Author: Armando Machado
Journal: Proc. Amer. Math. Soc. 84 (1982), 303-307
MSC: Primary 22E15; Secondary 58A05
DOI: https://doi.org/10.1090/S0002-9939-1982-0637188-2
MathSciNet review: 637188
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Abstract: We prove the following result. Let $ V$ be a smooth manifold and let $ G \subset V$ be a locally closed set with a group structure such that both multiplication and inversion are smooth maps; then $ G$ is an imbedded smooth submanifold of $ V$. This result is a generalization of the well-known fact that a closed subgroup of a Lie group is itself a Lie group, because we are not assuming any group structure in the manifold $ V$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0637188-2
Keywords: Closed subgroup of a Lie group, strictly tangent vector, flow-invariant set
Article copyright: © Copyright 1982 American Mathematical Society