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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]].\nA *Lie subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $H$ of $G$ together with a topology and a smooth structure on $H$ such that:\n\n1. $H$ with this structure is a Lie group;\n2. the inclusion map $H \\to G$ is an [[immersion|6RW68JGBHC]].\n\nIf moreover the inclusion is a homeomorphism onto $H$ with the subspace topology, then $H$ is an *embedded Lie subgroup*.\nEquivalently, $H$ is an embedded [[submanifold|VKFW27WXEZ]] of $G$.\n\n## Remarks\n\nExample: in the torus $T = S^1 \\times S^1$, let $\\alpha$ be irrational and $H = \\set{(e^{it}, e^{i\\alpha t}) : t \\in \\R}$, with the topology and smooth structure of $\\R$ transported by $t \\mapsto (e^{it}, e^{i\\alpha t})$.\nThis map is an injective homomorphism, since $\\alpha$ is irrational, and its derivative $(i e^{it}, i\\alpha e^{i\\alpha t})$ never vanishes, so it is an immersion.\nSo $H$ is a Lie subgroup, but not an embedded one: $H$ is dense in $T$.\n\nNon-example: the circle $C = \\set{z \\in \\C : \\abs{z} = 2}$ is an embedded submanifold of the Lie group $\\C^\\times$, but it is not a subgroup, since $2 \\cdot 2 = 4 \\notin C$.\n\nSome authors use \"Lie subgroup\" only for embedded Lie subgroups, and call the general ones immersed or virtual Lie subgroups.\nAn embedded Lie subgroup is always closed in $G$.\n",
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