Left Coset of Stabilizer in Group of Transformations

Theorem
Let $S$ be a non-empty set.

Let $G$ be a group of permutations of $S$.

Let $t \in G$.

Let $G_t$ be the set defined as:
 * $G_t = \set {g \in G: \map g t = t}$

Then each left coset of $G_t$ in $G$ consists of the elements of $G$ that map $t$ to some element of $S$.

Proof
Let $x \in G$.

Consider the left coset $x G_t$.

Let $\map x t = s$.

Then: