Definition

the set of all elements of the group that don't change a particular element in the space acted upon by the group.

Can be generalized to a stabilizer of a subset, where we require the subset is unchanged by the group action (though its elements may be permuted)

Lemma There is a bijection from $G /G_x \to G(x)$ that is from the quotient of $G$ by the stabilizer in $G$ of $x$, $G_x$, to the orbit of $x$, $G(x)$. This can be used to show Orbit-stabilizer theorem