Let $G$ be a Lie group acting smoothly and transitively on a smooth manifold $M$, and let $p \in M$. Suppose $G_p$ is a closed Lie subgroup and $G/G_p$ is equipped with its quotient smooth manifold structure. Then the map
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\begin{align*}
G/G_p &\to M \\
gG_p &\mapsto g \cdot p
\end{align*}