I'm going through the proof given for Orbit stabilizer theorem here, but, I am stuck at the point that there is a bijection from $G/ {\rm Stab}(x) \to{\rm Orb}(x)$. Just before, they found that stabilizer is a subgroup and the group action function $\phi: G \times X \to X$:
$$ \phi(g) = g \cdot x$$
Breaks injectivity when $ g \equiv h \mod \text{Stab}(x)$. How does one go from this to the idea that fixing injectivity means to consider domain as cosets rather than elements?
Best Answer
In general if $f: A \to B$ is surjective, then the preimages $f^{-1}(\{b\})$ for all $b \in B$ form a partition of $A$. [See this for instance.] This automatically produces a bijection from the collection of preimages $f^{-1}(\{b\})$ to the elements of $B$.
In your specific example where $\phi: G \to \text{Orb}(x)$, you can show that the preimages $\phi^{-1}(\{y\})$ for $y \in X$ happen to be the cosets of $\text{Stab}(x)$. This is the crux of the "$\phi(g) = \phi(h) \iff g^{-1} h \in \text{Stab}(x)$" claim in the middle of the proof.