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For a group action $F:G\times X\to X$ and fixed $x$, the orbit map $f_x(g)=F(g,x)$ hides the stabilizer subgroup $G_x$: $f_x(g)=f_x(h)$ exactly when $h^{-1}g\in G_x$, equivalently when $g$ and $h$ lie in the same left coset of $G_x$.

For a Hamiltonian group action, a regular value level preserved by a free proper group action has a smooth quotient. The restricted symplectic form descends to a symplectic form on that quotient when the kernel is exactly the orbit directions; this is symplectic reduction.

A group action on a set $X$ is transitive when every two points lie in the same orbit of a group action, equivalently when for all $x,y\in X$ there is a group element $g$ with $gx=y$.

The symmetric group acts on multivariate polynomials by replacing each variable $x_i$ by $x_{\sigma (i)}$. With $(\sigma \tau )(i)=\sigma (\tau (i))$, substitution gives $\sigma \cdot (\tau \cdot f)=(\sigma \tau )\cdot f$. Thus it is a left group action. A polynomial encoding a combinatorial partition has a group orbit encoding all relabelings and a stabiliser subgroup encoding its label symmetries.