Lie subgroup

Remarks

Example: 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 $t \mapsto (e^{it}, e^{i\alpha t})$. This 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. So $H$ is a Lie subgroup, but not an embedded one: $H$ is dense in $T$.

Non-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 $2 \cdot 2 = 4 \notin C$.

Some authors use "Lie subgroup" only for embedded Lie subgroups, and call the general ones immersed or virtual Lie subgroups. An embedded Lie subgroup is always closed in $G$.