Solution
= Solution
For a <group action> of $G$ on a set $X$ and $x\in X$, the <orbit-stabilizer theorem> states
$$
|Gx|=[G:G_x],\qquad G_x=\{g\in G:gx=x\}.
$$
If $G$ is finite, this becomes $\boxed{|G|=|Gx|\,|G_x|}$. The underlying bijection sends the left <coset> $gG_x$ to $gx$. Equality of two image points is equivalent to $g_2^{-1}g_1\in G_x$, so this map is both well-defined and injective, and it is surjective by the definition of the <group orbit>.