Given an manifold $X$ and a Lie group $G$ acting on $X$, then the action is called proper if given a compact set $K \subseteq X$ then $\{g\in G:gK\cap K\neq\emptyset\}$ has compact closure in $G$ (alternative definition at @sharpe2000differential page 145). The meaning (not obvious) is that the orbits don't get too horrible. For example, the action of $\mathbb{Z}$ on $\mathbb{S}^1$ given by irrational rotations
$$ n\cdot e^{i\theta}=e^{i(\theta+2\pi n \alpha)} $$with $0<\alpha<1$ irrational is not a proper action.
Proposition
Given a Lie group $G$ and a closed subgroup $H$ then the right action of $H$ on $G$ is proper.
$\blacksquare$
(see @sharpe2000differential Proposition 2.2 page 145).
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Author of the notes: Antonio J. Pan-Collantes
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